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Coplanar Anode Implementation in Compressed Xenon Ionization Chambers
by
Scott Douglas Kiff
A dissertation submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy (Nuclear Engineering and Radiological Sciences)
in the University of Michigan 2007
Doctoral Committee: Associate Professor Zhong He, Chair Professor Frederick D. Becchetti, Jr. Professor Ronald F. Fleming Associate Professor David K. Wehe
© Scott Douglas Kiff ___________________________
All rights reserved
2007
ii
ACKNOWLEDGEMENTS
I would first like to thank Prof. Zhong He for guiding my education over the past
four years. I have appreciated his willingness to help when I needed it most, allow me to
try new ideas when I had them, and help me grow as an engineer. I expect that I will still
be learning from him years from now.
Professors Franklyn Clikeman, Chan Choi, and Martin Lopez de Bertodano of
Purdue University have supported me for nearly a decade now, and for that I am most
grateful. Prof. Choi first encouraged me to pursue a Ph.D. when I did not believe I was
capable; Prof. Clikeman, Ed Merritt of Purdue, and Dr. Robert Bean of Idaho National
Laboratory introduced me to the wonderful worlds of radiation detection and nuclear
reactors. Prof. Bertodano guided me through the bumpy ride I call my master’s thesis—
and showed me how to apply the course material I had been learning to actual problems.
Professors Glenn Knoll and David Wehe of the University of Michigan have
taught me in courses and provided much technical insight and career guidance over the
past few years. In addition, they must be very pleased that I am finally graduating so that
I will stop asking for scholarship and employment reference letters, letters for which I
express my deepest gratitude. Along this line, Ed Birdsall and Peggy Gramer, also of the
University of Michigan, have provided countless hours of assistance with computers,
facilities, scholarship applications, and navigating Rackham’s sea of requirements.
Drs. Aleksey Bolotnikov, Royal Kessick, Pavel Kiselev, and Prof. Gary Tepper
have all been invaluable resources in my helping me fill my detectors and understand
experimental results. The National Science Foundation has provided me great flexibility
in pursuing my education via a Graduate Research Fellowship. I am appreciative of their
support and their willingness to continue to fund me after changing universities and
research topics not once, but twice.
iii
The members of my research group have always been very supportive and
helpful: Profs. Jim Baciak and Ling-Jian Meng; Drs. Clair Sullivan, Cari Lehner, Feng
Zhang, Dan Xu, and Keitaro Hitomi; soon-to-be Dr. Ben Sturm; Jim Berry; and Steve
Anderson, Miesher Rodrigues, Weiyi Wang, Yuefeng Zhu, Willy Kaye, and Chris Wahl.
Thanks for years of help on the Shockley-Ramo Theorem, Geant4, LabVIEW, Matlab,
peak-hold circuits, and just being good friends.
Dave Griesheimer, Heath Hanshaw, Troy Becker, Greg F. Davidson, and Jeremy
Conlin have provided countless instances of help with topics such as math and computer
programming, in addition to great conversations and good friendships. My experience at
Michigan would not have been half as enjoyable without them.
Will, Kirsten, Caitlin, and Mackenzie White (and now Victoria!) deserve special
mention in these acknowledgements. How could Natalie and I have survived Ann Arbor
without Will’s grilling expertise? Or brownies? Thanks for almost three years of letting
us spend weekends with you, decompressing, playing with Caitlin and Mackenzie, and
knowing exactly what best friends are.
Non-Wolverine friends have also been very supportive over the years: Mat
Mattox, Tony Waltman, and Jason Larson have been wonderful friends since high school,
and will continue to be for decades; Natalie Yonker, Josh Walter, Derek Hounshel, and
Travis Croy made long hours in the Nuclear Engineering Building very enjoyable.
Finally, I could not have completed this dissertation and the years of education
leading up to its culmination without the unconditional support of my family: my parents,
my sister Cori, Dennis and Ella, and Lindsay and Eric (and Emma!). Above all I would
like to thank Natalie, who has been by my side every step of the way for more than a
decade showing her love and support. These family members have truly helped me
through the periods of stress, shared with me in elation, and have worked overtime to try
to figure out exactly what nuclear engineering is.
iv
TABLE OF CONTENTS
ACKNOWLEDGEMENTS................................................................................................ ii
LIST OF FIGURES ......................................................................................................... viii
LIST OF TABLES............................................................................................................ xii
CHAPTER
1 INTRODUCTION .................................................................................................... 1
1.1 Properties of High-Pressure Xenon .................................................................... 1
1.2 Possible Applications for HPXe Ionization Chambers....................................... 3
1.3 Development of Gridded HPXe Ionization Chambers ....................................... 4
1.3.1 Planar Configuration................................................................................... 5
1.3.2 Cylindrical Configuration ........................................................................... 7
1.4 Alternatives to the Gridded HPXe Ionization Chamber ..................................... 8
1.4.1 Uncompensated Cylindrical Two-Electrode Devices................................. 9
1.4.2 Rise Time-Compensated Cylindrical Two-Electrode Devices ................. 10
1.4.3 Non-Standard Geometries......................................................................... 11
1.4.4 Multiple-Anode Chambers........................................................................ 12
1.4.5 Signal Noise Filtering ............................................................................... 15
1.5 Sensing the Gamma-Ray Interaction Position in HPXe ................................... 16
1.5.1 Position Sensing with Scintillation Time Stamps..................................... 17
1.5.2 Position Sensing with Electrode Signal Ratios......................................... 18
1.6 Objectives of This Work................................................................................... 18
2 SIGNAL INDUCTION THEORY ......................................................................... 20
2.1 Signal Induction on a Conductor by Moving Point Charges ............................ 20
2.2 Single-Polarity Charge Sensing ........................................................................ 22
2.2.1 Planar Two-Electrode Systems .................................................................. 22
2.2.2 The Frisch Grid .......................................................................................... 24
v
2.2.3 Coplanar Grid Anodes ............................................................................... 25
2.2.4 Position Sensing Using Coplanar Grid Anodes ......................................... 28
3 DETECTOR DESIGN............................................................................................ 32
3.1 Design Concept................................................................................................. 32
3.2 Electrostatic Potential Theory........................................................................... 35
3.2.1 General Potential Theory ........................................................................... 35
3.2.2 Weighting Potential Representation .......................................................... 41
3.3 Chamber Optimization...................................................................................... 43
3.3.1 Numerically Determining the Weighting Potential Fourier Coefficients .. 43
3.3.2 Charge Induction Uniformity Dependence Upon the Number of Wires ... 51
3.3.3 Electronic Noise as a Function of the Number of Anode Wires................ 53
3.3.4 The Optimal Balance of Electronic Noise and Weighting Potential ......... 55
3.3.5 The Importance of Signal Subtraction ....................................................... 59
3.3.6 Estimating the Critical Electrode Biasing of the HPXe Chamber ............. 60
4 INITIAL EXPERIMENTS ..................................................................................... 64
4.1 Detector Preparation and Gas Filling................................................................ 64
4.1.1 Detector Construction ................................................................................ 64
4.1.2 Gas Purification and Detector Filling ........................................................ 66
4.2 Experiments ...................................................................................................... 68
4.2.1 Capacitance Measurements........................................................................ 68
4.2.2 Electronic Noise Measurements ................................................................ 69
4.2.3 Preamplifier Waveforms............................................................................ 72
4.2.4 Cathode Biasing Spectra ............................................................................ 73
4.2.5 Anode Biasing Spectra............................................................................... 75
4.2.6 Effects of Background Radiation and Source Collimation........................ 79
4.2.7 Optimal Shaping Filter............................................................................... 82
4.2.8 System Linearity ........................................................................................ 83
4.2.9 Discussion of Preliminary Results ............................................................. 86
5 RADIAL POSITION SENSING ............................................................................ 88
5.1 Motivation for Position Sensing ....................................................................... 88
5.2 Coplanar Anode Position Sensing Theory........................................................ 89
vi
5.3 Simulations ....................................................................................................... 93
5.3.1 Maxwell 3D Electrostatic Simulations ...................................................... 94
5.3.2 Matlab Waveform Generation Code.......................................................... 95
5.3.3 Geant4 Monte Carlo Simulations............................................................... 98
5.3.4 Simulation Results ..................................................................................... 99
5.4 Experiments .................................................................................................... 106
6 CONSIDERATIONS FOR IMPROVING PERFORMANCE............................. 111
6.1 Energy Resolution Enhancement.................................................................... 111
6.1.1 Simulation Methods for Physical Process Contributions......................... 111
6.1.2 Spectral Contributions in Pure Xe with a 20 μs Shaping Time ............... 114
6.1.3 Analysis of Events Along an Arc of Fixed Radius .................................. 123
6.1.4 Photopeak Broadening Contributions for Realistic Shaping Times ........ 126
6.2 The Effects of Multiple-Site Events and Interaction Location ....................... 128
6.3 Structural Material Effects on the Energy Spectrum ...................................... 137
6.4 Summary of Simulation Results ..................................................................... 138
7 IMPROVING ENERGY RESOLUTION WITH COOLING ADMIXTURES... 140
7.1 Increasing Drift Velocity with Cooling Admixtures ...................................... 140
7.1.1 The Need for Larger Electron Drift Velocities ........................................ 140
7.1.2 The Effects of Cooling Admixtures......................................................... 140
7.1.3 Effects of Cooling Admixtures on Other Detector Properties ................. 143
7.2 Simulation Results for Xenon+Hydrogen Mixtures ....................................... 144
7.3 Hydrogen Addition and Gas Filling................................................................ 147
7.3.1 Choosing the Optimal Hydrogen Concentration...................................... 147
7.3.2 Gas Mixing, Purification, and Filling ...................................................... 148
7.4 Gamma-Ray Measurements with Xenon+Hydrogen Gas Mixtures ............... 149
7.4.1 Initial Experiments................................................................................... 149
7.4.2 Radial Position-Sensing Experiments...................................................... 150
8 CONCLUSIONS................................................................................................... 158
8.1 Summary of Pure Xenon Experiments and Simulations................................. 158
8.2 Summary of Xenon+Hydrogen Experiments and Simulations....................... 160
8.3 Suggestions for Future Work .......................................................................... 161
vii
REFERENCES ............................................................................................................... 163
viii
LIST OF FIGURES
Figure 1.1. A typical Frisch Grid with 4 wires/mm pitch...................................................4
Figure 1.2. A parallel-plate gridded HPXe chamber. .........................................................6
Figure 1.3. A typical cylindrical gridded HPXe chamber. .................................................8
Figure 1.4. A hemispherical HPXe chamber. ...................................................................12
Figure 1.5. The first coplanar anodes implemented in HPXe...........................................13
Figure 1.6. A schematic of the DACIC concept with two symmetric anode wires. .........14
Figure 1.7. Position sensing along the X coordinate (as shown). .....................................17
Figure 2.1. An illustration of important parameters in the Shockley-Ramo Theorem. ....20
Figure 2.2. A schematic of a gridded ionization chamber (top). ......................................25
Figure 2.3. A coplanar grid anode with cathode 1............................................................26
Figure 2.4. A 137Cs depth-separated energy spectrum in coplanar-grid CdZnTe (top). ...31
Figure 3.1. Cross-sectional schematics of the proposed detector showing end (left).......34
Figure 3.2. Comparison of one period of the simulated weighting potential difference and the approximation along the solution interface for 2, 4, and 12 anode wires. ....45
Figure 3.3. Comparison of the Maxwell 3D simulated anode weighting potential difference and the first 25 terms of the Fourier series for 2, 4, and 12 wires. ...........46
Figure 3.4. Demonstrating the convergence of the Fourier series for (a) 2 anode wires..48
Figure 3.5. (a) Comparing the simulated and calculated ( )20 mm,diffϕ θ . ......................49
Figure 3.6. Axial variations in the anode weighting potential difference for 2 to 16 anode wires along the line ( ) ( ), 20 mm,0r θ = . ........................................................50
Figure 3.7. Axial variations in the anode weighting potential difference for 8 anode wires along lines with 10θ = ° ...................................................................................51
Figure 3.8. The dependence of ( ), 0, 0diff r zϕ θ = = on the number of anode wires. .......52
Figure 3.9. The expected A250 noise as a function of detector capacitance and FET. ....53
Figure 3.10. Simulated 137Cs energy spectra for different anode wire numbers (top)......58
Figure 3.11. The simulated anode weighting potential difference compared to the weighting potential of the collecting anode only.......................................................59
ix
Figure 3.12. A comparison of simulated 137Cs energy spectra for a 12-wire anode when signal subtraction is employed versus when only the collecting anode signal is utilized....................................................................................................................60
Figure 4.1. A photograph of the anode structure in a test assembly. ................................65
Figure 4.2. A photograph of the detector prior to final assembly.....................................65
Figure 4.3. A photograph of the Macor spacer (center, white).........................................66
Figure 4.4. A typical HPXe purification and filling station using a spark purifier...........67
Figure 4.5. A diagram of the capacitance measurement concept. ....................................68
Figure 4.6. Measured ENC (top).......................................................................................71
Figure 4.7. Measured collecting (green) and noncollecting (blue) anode waveforms......72
Figure 4.8. A block drawing of the equipment setup for gamma-ray experiments. .........73
Figure 4.9. 137Cs gamma-ray spectra as a function of the applied cathode bias when both anodes are grounded. .........................................................................................74
Figure 4.10. 137Cs spectra as a function of collecting anode bias with the noncollecting anode grounded and the cathode held at -4000 V......................................................76
Figure 4.11. A preamplifier waveform from the noncollecting anode (yellow)...............77
Figure 4.12. The number of positive noncollecting anode counts as a function of the collecting anode bias; the source is 137Cs, and the cathode is held at -4000 V..........78
Figure 4.13. The effects of background correction (red) and source collimation (blue) ..79
Figure 4.14. The detector response with the source collimated upon the detector center (black) and the opposing end gas spaces where the anode (red) ...............................80
Figure 4.15. The change in the photopeak region for a selection of Gaussian filters using shaping times from 8 to 16 μs. .........................................................................81
Figure 4.16. The optimized 137Cs energy spectrum exhibits a 6.0% FWHM energy resolution and greatly reduced counts in the lower part of the Compton continuum. .................................................................................................................82
Figure 4.17. A spectrum collected with 133Ba, 137Cs, and 60Co gamma-ray sources. .......84
Figure 4.18. The measured photopeak centroid plotted against the true gamma-ray energy (black) ............................................................................................................85
Figure 5.1. Geant4 simulated distribution of the charge cloud diameter in the HPXe detector for 662-keV depositions...............................................................................96
Figure 5.2. Simulated effect of shaping time choice on the photopeak region...............101
Figure 5.3. The calculated interaction radius plotted against the true interaction radius for several shaping time constants. ..........................................................................102
Figure 5.4. The radially-separated energy spectrum for the 137Cs simulation. ...............104
Figure 5.5. The effect of radial corrections on the simulated 137Cs energy spectrum. ...105
x
Figure 5.6. A connection diagram of the detection system used for radial position sensing measurements..............................................................................................106
Figure 5.7. A radially-separated experimental 137Cs energy spectrum...........................108
Figure 5.8. An experimental 137Cs radially-separated spectrum with background stripped out, but otherwise identical to Figure 5.7...................................................108
Figure 5.9. The effect of radial corrections on the experimental 137Cs data. ..................110
Figure 6.1. The response of the shaping filter to events at (18 mm, 0, 0) ......................113
Figure 6.2. A comparison of the true deposited energy spectrum to the spectrum broadened by Fano carrier statistics.........................................................................115
Figure 6.3. A comparison of the energy spectrum broadened by Fano statistics and the weighting potential distribution to the spectra when charge recombination ...........116
Figure 6.4. Examining the electric field lines along the central plane near the anodes when Vcat = -4000 V; collecting anode wires are surrounded by red. .....................117
Figure 6.5. Examining the effects of shaping upon the energy spectrum.......................119
Figure 6.6. Comparing the shaped spectrum without noise............................................120
Figure 6.7. A comparison of the simulation results when the field distribution projected throughout the entire volume corresponds to the planes z = 0 or z = 48 mm. ..........................................................................................................................121
Figure 6.8. A comparison of the FWHM attributed to each physical process studied in the simulations. ........................................................................................................122
Figure 6.9. Investigating the effect of azimuthal angle on the distribution of signal amplitudes. ...............................................................................................................124
Figure 6.10. A plot of preamplifier and shaped waveforms for different azimuthal angles (left). .............................................................................................................125
Figure 6.11. A comparison of the peak broadening terms for two different shaping times.........................................................................................................................127
Figure 6.12. Simulated 137Cs energy spectra plotted as a function of the number of interaction sites. .......................................................................................................131
Figure 6.13. A comparison of simulated 137Cs radially-separated energy spectra for (a) 1-site, (b) 2-site, (c) 3-site, and (d) 4-site events. ....................................................132
Figure 6.14. Simulated 137Cs spectra sorted by the number of interaction sites. ............133
Figure 6.15. Maxwell 3D simulation results displaying the electric field strength on the XZ plane when the cathode and collecting anode are biased to -4000 V and +1400 V, respectively. .............................................................................................135
Figure 6.16. Simulation results of 137Cs spectra for all events (black) ...........................136
Figure 6.17. The effect of replacing structural materials in the HPXe detector with vacuum on simulated 137Cs energy spectra..............................................................138
xi
Figure 7.1. The momentum-transfer cross section for Xe (reprinted from [95])............141
Figure 7.2. Electron drift speed (mm/μs)........................................................................143
Figure 7.3. A comparison of w in doped HPXe, where W0=21.9 eV. ............................144
Figure 7.4. A comparison of detector responses for full-energy events when several shaping time constants are used...............................................................................145
Figure 7.5. A comparison of the simulated distribution of pulse amplitudes after recombination for two gas compositions. ................................................................146
Figure 7.6. A comparison of simulated 137Cs spectra using a Xe+H2 gas mixture before and after photopeak alignment......................................................................147
Figure 7.7. Experimental 137Cs energy spectra as a function of radial bin. ....................151
Figure 7.8. A collimated 137Cs raw spectrum compared to the background-stripped and the aligned results; a test pulse appears near channel 800. ......................................152
Figure 7.9. A comparison of 137Cs spectra with three different collimator configurations. .........................................................................................................153
Figure 7.10. Collimated source spectra after photopeak alignment................................154
Figure 7.11. A plot of the experimentally-measured photopeak centroids vs. the published gamma-ray energy...................................................................................156
Figure 7.12. A plot of the natural logarithm of the measured intrinsic resolution as a function of the natural logarithm of the normalized gamma-ray energy. ................156
xii
LIST OF TABLES
Table 1.1. A comparison of common radiation detection media........................................2
Table 3.1. A summary of fixed design parameters. ..........................................................34
Table 3.2. Boundary conditions for the various weighting potentials. .............................42
Table 3.3. Values of α used to calculate the weighting potential difference in Figure 3.2...............................................................................................................................45
Table 3.4. Amptek A250 electronic noise contribution as a function of anode geometry. ...................................................................................................................55
Table 4.1. Measured capacitance values for the HPXe detectors. ....................................69
Table 4.2. Subtracted signal rise time vs. cathode bias while the anodes are grounded...73
Table 4.3. Sources used in the linearity measurement along with photopeak properties. 84
Table 4.4. The departure of measured peak centroids from the least-squares solution. ...85
Table 6.1. A list of important parameters in the energy resolution study.......................114
Table 6.2. A summary of important results from the physical effects study. .................123
Table 6.3. A comparison of FWHM contributions for two shaping times. ....................127
Table 6.4. Comparing spectral features as a function of the number of interaction sites. .........................................................................................................................131
Table 6.5. Comparing the simulated energy resolution before and after peak alignment..................................................................................................................134
Table 6.6. Comparing the prominence of event sequences for 137Cs gamma-ray detection. ..................................................................................................................134
Table 7.1. Gas properties of the filled HPXe detectors. .................................................149
Table 7.2. A summary of collimated 137Cs measurements during the shaping time study. ........................................................................................................................150
Table 7.3. A comparison of measured photopeak centroids as a function of radial bin.151
Table 7.4. A comparison of spectral parameters in the 137Cs collimation experiment. ..154
Table 7.5. Collimated source photopeak data for detector HPXe2.................................155
1
CHAPTER 1
INTRODUCTION
1.1 Properties of High-Pressure Xenon
High-pressure xenon (HPXe) gas is an attractive gamma-ray detection medium
due to several of its physical and nuclear properties. HPXe has a large detection
efficiency for gamma ray energies on the order of 100s of keV, due to its large atomic
number (Z = 54), which translates into high photoelectric absorption and Compton
scattering cross sections; in addition, the detection efficiency of these devices is enhanced
by the ability to use a high density of up to about 0.6 g/cm3 [1], coupled with a large
sensitive volume (commonly on the order of liters). The mean energy to produce an
ionization in HPXe, commonly referred to as w, is relatively low for a gas, less than 21
eV per electron-ion pair [1]. In addition, the Fano factor for HPXe is quite good,
measured near 0.13 ± 0.1 [2]. These two parameters combine to give HPXe ionization
chambers an attractive theoretical energy resolution of about 0.5% full-width at half-
maximum (FWHM) at 662 keV, the energy corresponding to gamma rays emitted from a 137Cs source.
There are other attractive qualities of HPXe as well. For example, HPXe
ionization chambers are ideal for use in uncontrolled environments, as detector response
has been shown to be uniform over large temperature ranges (20°C to 170°C) [3]; this is
postulated to be due to the relatively large ionization energy of xenon, which makes
thermal excitation a negligible factor [4]. HPXe ionization chambers can be operated at
room temperature, so there is no need for providing a cooling source. Unlike solid
detection media that derive their radiation detection capabilities from their crystal
structure, HPXe performance is not degraded by high radiation fluences, although
2
exposure to a significant neutron flux can activate the xenon. The primary isotopes
responsible for this induced activity are 129Xe and 131Xe, which form metastable states
that decay with half-lives on the order of ten days via transitions of 236 and 164 keV,
respectively [5, 6]. As a final point, the cost of HPXe gas is relatively inexpensive,
around $1/g, resulting in a cost per detector on the order of 100s of U.S. dollars [7].
Radiation interactions in HPXe create many excited Xe2 molecules; these
molecules return to the ground state by emitting detectable scintillation light [8-10].
Ionized atoms that recombine at the ionization site will also create scintillation. This
scintillation light can be useful in marking gamma-ray interactions for timing purposes.
Also, electrons accelerated through high electric fields in Xe also cause scintillation light
emission, and this light can be collected by a photomultiplier tube for spectroscopic
purposes. For this to occur, the ratio of the electric field magnitude to the atom density of
the gas must be greater than the scintillation threshold of 2.9x10-17 V/cm2 [11].
A summary of important properties of compressed xenon gas is listed in Table
1.1, along with a comparison to other common detection media [8, 12]. In this table,
common thicknesses of the media being compared to HPXe are given, along with an
equivalent thickness of HPXe that counts the same number of particles per unit area for a
planar gamma ray source emitting photons of energy 662 keV. This measure probably
underestimates the true efficiency of HPXe by neglecting the larger area of the detector.
Table 1.1. A comparison of common radiation detection media.
Medium Density (g/cm3)
Atomic numbers
w (eV/ion pair)
Thickness (cm)
Equivalent HPXe (cm)
NaI:Tl 3.671 11, 53 ~100 5.08 37.52
Ge 5.333 32 2.983 2 20.22
Cd0.8Zn0.2Te 63 48, 30, 52 5.03 1 11.82
HgI2 6.43 80, 53 4.33 1 14.92
HPXe 0.5 54 21.9 ― ―
1 From G.F. Knoll, Radiation Detection and Measurement, 3rd ed., Table 8.3. 2 Photon attenuation data is from the NIST XCOM database. 3 From G.F. Knoll, Radiation Detection and Measurement, 3rd ed., Table 13.3.
3
1.2 Possible Applications for HPXe Ionization Chambers
HPXe is an attractive detection medium for applications that require a detector to
have better energy resolution than can be provided by scintillators, better efficiency than
semiconductors can offer, and the ability to operate without cooling and over a wide
range of temperatures without sacrificing performance. One of the applications of
greatest interest is detecting special nuclear materials (SNM) at borders and ports.
Detecting SNM requires sensitivity over a large energy range, from 235U gamma rays
with energies between 150 and 200 keV up through 2614 keV, a gamma emitted by the 232Th and 232U daughter product 208Tl [13]. The energy resolution must be good enough
such that nearby gamma lines do not interfere in measuring the lines of interest; for
example, the 235U 186-keV complex should be distinguishable from the 238-keV line
originating from U and Th daughter products [14]. HPXe chambers have been identified
as a medium that can meet the sensitivity and resolution needs for SNM monitoring.
A second application of great interest for HPXe is environmental monitoring of
radioactive soils, which can demand good spectroscopic performance while lowering a
detector into a borehole up to depths of 250 feet [15]. Radioisotopes of interest in these
surveys typically emit gammas with energies on the order of 1 MeV, and a detector must
be sensitive enough to measure contaminant activities near the naturally-occurring
background level, while rugged enough to withstand dose rates near 1 krad/hr in highly
contaminated soils. Temperatures in deep wells can rise above 100°C, complicating the
measurement [16]. HPXe chambers can provide the combination of sensitivity,
temperature stability, and energy resolution needed for these activities.
There has been interest in using HPXe ionization chambers for basic science as
well. Researchers have been attracted by the combination of high efficiency, good
energy resolution, and radiation hardness offered by HPXe, sending multiple HPXe
detectors into orbit aboard the MIR space station to study gamma-ray bursts [17, 18].
Another interesting proposal for HPXe chambers is to assist in the investigation of
neutrinoless double beta decay, a study which can help to understand the neutrino and
quantify its mass [19]. 136Xe, which constitutes 8.9% of natural Xe, undergoes the
2β(0ν) transition with the large decay energy of 2.48 MeV. Because HPXe ionization
chambers contain large quantities of gas, are simple to build and operate, have good
4
energy resolution near 2.5 MeV, and are the source of these decays, they have been
explored by two different groups for 2β(0ν) experiments [19, 20].
1.3 Development of Gridded HPXe Ionization Chambers
The first attempts at using HPXe ionization chambers in the early 1980s
employed ionization chambers with Frisch grids. The purpose of these grids is to shield
the anode from signal generation when charges drift through the bulk of the detector; a
detailed discussion will follow in Chapter 2 of this dissertation. Electrostatically
shielding the anode from moving charge in a large portion of the detector results in a net
anode signal that is generally only a function of the number of electrons collected by the
anode. A Frisch grid is typically formed from several small-diameter wires (10s to 100s
of μm); many of these wires are stretched to form a parallel array or a crossed grid, which
is placed near the anode. A crossed Frisch grid structure is shown in Figure 1.1 [15].
Wire separation is often an order of magnitude larger than the individual wire diameter.
The grid is held at a potential that: (i) creates a sufficient electric field in the ionization
region, which is between the cathode and the Frisch grid, to minimize electron
attachment; and (ii) minimizes charge collection on the Frisch grid itself. To accomplish
the latter goal, the electric field magnitude between the Frisch grid and the anode
typically must be significantly greater than the field magnitude in the ionization region
[21].
Figure 1.1. A typical Frisch Grid with 4 wires/mm pitch.
5
Gridded ionization chambers can be found in two distinct geometries: planar and
cylindrical. Planar configurations are convenient due to the uniform axial electric field; it
will be shown later that the charge created at the site of the initial interaction recombines
at a rate that is a function of the electric field. Thus, recombination effects can be made
uniform throughout the device. Cylindrical chambers are advantageous because of the
inherent lower capacitance this geometry affords, the potential for a balanced tradeoff of
charge induction and charge recombination as a function of radial interaction location,
and better utilization of the xenon gas space.
1.3.1 Planar Configuration
The first reported measurements using a HPXe ionization chamber were done
with a planar configuration [22, 23]. The HPXe ionization chamber incorporated a dual
drift region, utilizing a central anode plane with sensitive volumes on either side,
accompanied by a cathode/Frisch grid set on each side of the anode plane. This geometry
creates a uniform electric field, which is maintained near the periphery via the use of
several drift electrodes. Each half of the active volume had a 93-mm diameter and a 38-
mm height. The gas density was 0.2 g/cm3, and the gas was doped with a small amount
of H2 gas (0.5%). The incorporation of a small admixture of H2 will be discussed in more
detail in Chapter 7; the purpose is to greatly increase the drift velocity of electrons with
only a small effect on other gas properties. The reported energy resolution measured
with this chamber was 2.7% FWHM for a 137Cs gamma-ray source. By measuring the
pulse amplitude as a function of the applied drift field and extrapolating to the infinite
field case, w of HPXe was measured with this system to vary from (21.4 ± 0.3) eV/ion
pair to (20.8 ± 0.3) eV/ion pair as the gas pressure was increased from 1.5 to 4.1 MPa
[23].
This geometry was also used in several other investigations [5, 6, 17, 19, 24-32].
A significant limitation of the Frisch grid was revealed by measurements using a dual-
drift-region HPXe chamber aboard the MIR space station: the measured energy
resolution degraded from 2.0% FWHM at 1 MeV during pre-flight testing to between 3
and 4% during flight, a change attributed to microphonic vibrations aboard MIR [26].
6
Using another planar configuration, Bolotnikov et al. studied the effects of gas
pressure on the measured energy resolution of HPXe ionization chambers [33]. The
geometry used was very basic, with a single anode, cathode, and Frisch grid forming a
single drift region. In this case, a 207Bi source was implanted on the surface of the
cathode; 207Bi has a relatively complex energy spectrum due to its multiple modes of
radioactive decay [34]. Increasing the Xe gas density had no measurable effect on the
energy resolution below 0.6 g/cm3, as the resolution was only a function of charge carrier
statistics. However, above a density of 0.6 g/cm3 the energy resolution degraded rather
quickly. After ruling out other possible sources of this degradation, the reason was found
to be ion-electron recombination along the δ-ray tracks in the ionization clouds. This
physical limitation on energy resolution is responsible for the optimal HPXe density
being just underneath this threshold of 0.6 g/cm3. This result was verified by Levin,
Germani and Markey [2, 20], who concluded that the intrinsic energy resolution is not
governed by Poisson statistics at high pressures.
Figure 1.2. A parallel-plate gridded HPXe chamber. The vertical columns are supporting (from top) the cathode, two field rings, the Frisch grid (white border), and the anode. The cathode and anode are separated by 6.0 cm; the sensitive volume’s diameter is 6.4 cm.
7
Finally, Mahler et al. [35] used a single-drift-region HPXe ionization chamber to
produce a spectrometer for field use with a sensitive volume of 160 cm3; this detector is
shown in Figure 1.2. With a collimated source in a lab setting, this device achieved 2%
FWHM resolution at 662 keV, which is the best reported resolution to date in a planar
HPXe ionization chamber. By maintaining the high voltage across capacitors and
charging the system for 10 seconds every 30 minutes using batteries, a portable device
was created that measured 2.5% FWHM at 662 keV without source collimation. Biasing
the detector in this fashion reduced peak broadening from high-voltage ripple.
1.3.2 Cylindrical Configuration
Ulin et al. [36, 37] desired a HPXe chamber with high detection efficiency and
good energy resolution. To achieve these goals, a cylindrical geometry utilizing a Frisch
grid was developed; in the cylindrical geometry, the sensitive volume is a larger fraction
of the total gas space, and the Frisch grid makes charge induction on the anode more
uniform as a function of radial position. This detector’s sensitive region was 36 cm in
length and 8.3 cm diameter, a sensitive volume of nearly 2 liters. The chamber was filled
with Xe+0.2%H2 for improved electron drifting, limiting the total electron drift time to
less than 10 μs. The measured energy resolution for a 137Cs gamma source was 4.0%
FWHM, including electronic noise. Nearly identical HPXe chambers were developed [4-
6, 13, 28, 31, 32, 38-41], with Ulin et al. achieving 2.9% FWHM at 662 keV using a
HPXe volume of just over 5 liters [38].
The next notable set of improvements in cylindrical gridded HPXe detectors
focused on reducing Frisch grid microphonics [29, 30, 42, 43]. In this study the overall
detector dimensions and fill gas were nearly identical to the previous study, but the grid
was formed from a metallic foil of thickness 0.25 mm, into which a regular pattern of 5-
mm by 3-mm holes was introduced for transparency to drifting electrons; the rectangular
holes were separated by 0.25 mm in either direction. This is a sturdier design than an
array of individual wires, and vibrational issues were expected to diminish via this
design. In addition, the grid was not just held in place at each end, but also at two
intermediate locations using ceramic insulators. This feature further dampened
microphonic vibrations. The total energy resolution at 662 keV was measured as 2.4%
8
and 2.2% FWHM for irradiation of the chamber side wall and end, respectively.
Bolotnikov and Ramsey investigated a similar mesh design at about the same time, with
nearly identical energy resolution measurements [44, 45]. The development of sturdier
shielding grids led to many subsequent investigations [14, 15, 31, 32, 46-48]. A
schematic of a typical cylindrical gridded chamber is shown in Figure 1.3; this chamber
includes an electroformed mesh grid with periodic spacers for microphonic reduction
[49].
Figure 1.3. A typical cylindrical gridded HPXe chamber. The cathode is not shown.
1.4 Alternatives to the Gridded HPXe Ionization Chamber
Obviously the energy resolution using gridded HPXe chambers can be quite good,
as shown in the previous section. Often, good chambers will obtain total energy
resolution values near 2% FWHM at 662 keV for small-dimension chambers (drift
distances of around 2 cm), or between 3-4% FWHM for larger chambers (drift distances
of about 5 cm). Although the measured resolution is superior to most scintillators, and is
approaching the performance of good room-temperature semiconductor devices, there is
still a large gap between the measured performance and the theoretical energy resolution
of 0.5% FWHM at 662 keV. It is commonly thought that microphonic vibration of the
Frisch grid plays a major role in resolution degradation; this effect was measured using a
planar gridded detector and found to be significant [26].
9
In addition, design, construction, and operational problems can be associated with
Frisch grids. The grid wire diameter, spacing, and operating potential must be chosen
correctly: the grid must effectively shield the anode from electron motion in the
ionization region, while being transparent to electrons so that as many as possible pass
through the grid and generate a signal on the anode. Assuming the anode is grounded,
the optimal grid bias is a function of the bias applied to the cathode due to electrostatics,
as well as the gas density because of charge recombination arguments (i.e., the optimal
grid bias will balance any spatial variation in charge recombination with the shielding
inefficiency of the grid). Finally, Frisch grids can be difficult to manufacture due to their
delicate wires, the need for exact geometry, and the need for proper tensioning of each
wire.
Given these drawbacks to Frisch grid use, much effort has been spent on finding a
viable functional alternative. Some efforts have focused on simple two-electrode devices
due to their simplicity; some try to use pulse height compensation based upon electron
drift time; some use more complex anode structures that function as both the signal
electrode and the shielding grid; and there have also been efforts to use post-
measurement data processing to filter out the effects of grid vibrations. Each technique
will be discussed in more detail in this section, along with its inherent advantages and
disadvantages.
1.4.1 Uncompensated Cylindrical Two-Electrode Devices
Two-electrode devices, which are the simplest detectors because they have only a
single anode and a single cathode, were among the earliest HPXe detectors developed.
Because there is no Frisch grid, microphonic degradation of energy measurements should
be minimized, and the reduced detector capacitance should improve electronic noise. On
the other hand, the lack of a Frisch grid means the pulse amplitude will be a function of
the gamma’s interaction location, but this can be somewhat improved by using a
cylindrical geometry.
Dmitrenko et al. reported results on a two-electrode device in 1981 [50]. This
detector was expected to exhibit energy resolution of 5.4% FWHM, calculated by
considering a uniform distribution of photoelectric absorptions and the radial distribution
10
of the induced charge on the anode. Using a gas mixture including 0.3% H2, the
measured energy resolution was 6% FWHM for a 137Cs gamma-ray source. Other
published experiments have been based upon this initial design [3, 4, 16, 28-32, 41, 51-
53].
The best result in this geometry was reported by Dmitrenko et al. [3], and
separately by Ulin et al. [16]. This device had a measured energy resolution of 4%
FWHM at 662 keV. In addition, a temperature study was done in which the detector
response was measured from 20ºC to 170ºC; the maximum temperature was determined
by the integrity of the pressure vessel under increasing gas pressure. This study found no
measurable shift in photopeak centroid, and the energy resolution remained unchanged
within experimental error.
1.4.2 Rise Time-Compensated Cylindrical Two-Electrode Devices
The drawback of uncompensated cylindrical two-electrode devices is that there is
a significant dependence of the signal amplitude upon the gamma-ray interaction
location. To remove this detrimental effect, the pulse rise time can be measured and used
to separate events into small bins based upon electron drift distance. Pulse height
normalization can then be applied to each bin.
This technique was first reported by Bolotnikov and Ramsey [44, 45]. A HPXe
chamber of diameter 4.8 cm was used, and events with rise times measured between 14
and 16 μs were accepted. This rise time envelope corresponded to events near the
cathode, and resulted in an energy spectrum with a measured resolution of 2.5% FWHM
at 662 keV. For comparison, without rise time selection the energy spectrum of all
events in the device had a measured resolution of about 8% FWHM. A similar attempt
by Troyer, Keele, and Tepper had less-satisfactory results [54]. Smith, McKigney, and
Beyerle measured the rise time of each event and implemented a correction factor derived
from simulations to correct the pulse amplitude [55].
Measuring the rise time of the anode’s preamplifier signal can introduce a
significant amount of uncertainty into the compensation process. Because electrons drift
quite slowly through Xe or even Xe+H2 mixtures, and generally the noise fluctuations are
fairly significant when compared to the signal amplitude, good estimations of the start
11
and stop time can be difficult to make. One alternative is to use scintillation light as the
start and stop signals to measure electron drift time. There are a few technical challenges
in this method [56]: coupling a photomultiplier tube to a high-pressure chamber is
difficult; the emitted scintillation light is in the ultraviolet range, meaning wavelength
shifters or UV-sensitive photocathodes must be used; and finally the number of
scintillation photons emitted is quite small, so the system must be sensitive to a small
signal.
Lacy et al. employed scintillation light collection for electron drift time
measurement in a cylindrical HPXe ionization chamber [11]. A light guide was mated to
a transparent window at one end of the chamber, and a photomultiplier tube was coupled
to this light guide. Scintillation light created from de-excitation and recombination at the
ionization site marked the start time of the signal. As a result of a very large electric field
near the anode surface—over 50 kV/cm—electrons drifting near the anode transferred
enough energy in collisions to create excitations that emitted scintillation light upon
returning to ground state. This secondary scintillation marked the end of electron drift.
By correcting the anode pulse amplitude according to the measured drift time, the energy
spectrum for 22Na improved from essentially no 511 keV photopeak without pulse-height
correction to 2.3% FWHM when the correction was employed.
1.4.3 Non-Standard Geometries
One of the earliest attempts at using HPXe ionization chambers had a geometry
akin to the familiar silicon drift detector [57]. In this case, a chamber was created
between two plates separated by 1.5 cm. Distributed along the plates were conducting
strips placed at regular intervals and biased in uniform steps. An anode wire was placed
midway between the two plates near one end of the detection volume. The electrostatic
field created in this detector focused any electrons onto the anode, and the geometry
removed nearly all position dependence of the anode signal due to the excellent
electrostatic shielding provided by the field electrodes. The measured energy resolution
was 3.1% FWHM using a 137Cs gamma-ray source. Because of the small width of this
chamber, a significant x-ray escape peak was measured. A similar principle was
employed by Bolotnikov et al., who implemented a Frisch collar design in HPXe [58].
12
The drawback of these concepts is that charge induction requirements prevent the use of
large-volume chambers.
Figure 1.4. A hemispherical HPXe chamber. The cathode diameter is 3.5 inches.
One interesting attempt at using a novel geometry, shown in Figure 1.4, was
reported by Kessick and Tepper [59]. This detector employed a hemispherical geometry,
which is theoretically superior to the cylindrical two-electrode chamber in terms of
charge-induction uniformity throughout the entire device: this is because in a cylinder the
charge-induction deficit will go as ( )ln r , whereas in a spherical geometry the deficit
goes as 1 r , which is more ideal. The source of this advantage ends up being the
Achilles’ heel of the hemispherical detector: the electric field goes as 21 r , which means
that an extremely large bias must be applied across the detector to obtain an acceptably
small recombination rate and a uniform drift velocity throughout the detector. The
measured energy resolution for a 137Cs spectrum was 6% FWHM.
1.4.4 Multiple-Anode Chambers
Another category of detectors that can be used to replace the Frisch grid in HPXe
ionization chambers is multiple-anode chambers. In the designs reviewed, the presence
of more than one anode channel allows the anode to function both as the readout channel
and as the Frisch grid, since the recorded anode signal is largely independent of charge
13
movement in most of the detection volume. These designs tend to be relatively
insensitive to microphonics.
The first attempts at incorporating multiple-anode structures into HPXe chambers
were in the Ph.D. dissertation of Sullivan [60, 61]. These detectors implement coplanar
anodes, which are shown in Figure 1.5. Coplanar anode theory will be discussed in great
detail in Chapter 2, so only a brief description will be given now. Coplanar anodes are
two symmetric anodes that are operated with a potential bias applied between them:
electrons are thus preferentially collected on one anode, deemed the collecting anode.
Because of the charge induction properties of this device, the collecting anode’s induced
charge less that on the noncollecting anode gives a resulting signal that is functionally
equivalent to the electrostatic shielding of a Frisch grid. Therefore, if the anode structure
is resistant to microphonics, a vibration-resistant alternative to the Frisch grid can be
employed. The major disadvantage to coplanar-anode HPXe chambers is that since there
are two readout channels and thus two preamplifiers, the net subtracted signal suffers
from increased electronic noise due to the presence of two noise sources instead of just
one.
Figure 1.5. The first coplanar anodes implemented in HPXe. (Left) The helical anode. (Right) The spiral anode and the back of the cathode.
The first coplanar anode design incorporated two wires wound around an
insulating rod, forming a double-helical anode structure. This rod formed the central axis
in a cylindrical HPXe ionization chamber. The best energy resolution obtained with this
14
device was 9% FWHM using a 137Cs point source, which is quite poor: this was
postulated to result from incomplete charge collection on the collecting anode. A second
anode design incorporated two spiraling anode strips metallized on a flat ceramic plate;
this anode was used in a parallel-plate HPXe chamber. The best energy resolution
obtained from this detector with a 137Cs source was 12.7% FWHM.
Figure 1.6. A schematic of the DACIC concept with two symmetric anode wires.
A concept similar to coplanar anodes was introduced by Bolotnikov et al. [58,
62], termed the dual-anode cylindrical ionization chamber, or DACIC. In the DACIC
two symmetric anode wires are stretched axially in a cylindrical HPXe ionization
chamber; these anode wires are each displaced from the central axis by a few millimeters.
A schematic of the DACIC is shown in Figure 1.6. These wires electrostatically shield
one another, thereby functioning as the Frisch grid; unlike the Frisch grid, the wires can
be made relatively large to dampen microphonic vibration. The DACIC concept has been
used in subsequent investigations [63, 64]. The DACIC has been operated in two modes:
quasi-coplanar readout and individual wire readout.
Quasi-coplanar operation refers to the fact that although one anode signal is
subtracted from the other, there is no potential bias applied between the anode wires, as
traditional coplanar theory requires: thus, collection on only one anode wire cannot be
15
ensured. Besides the potential for charge sharing, the charge induction on the two anodes
is not uniform in the anode plane, contributing to unnecessary peak broadening. Finally,
if an incoming gamma ray creates energy depositions on opposite sides of the chamber,
those two charge clouds will be collected by opposing wires, and the subtraction step will
greatly reduce the recorded energy. Given these facts it is not surprising that the energy
resolution was 4% FWHM at 662 keV, which is not as good as gridded chambers can
achieve.
Using individual wire readout introduces a tradeoff: because the final signal
includes only one noise source instead of the combination of two independent noise
sources, electronic noise is expected to be lower. However, the charge induction is
expected to be even less ideal, although the lack of signal subtraction removes problems
related to signal amplitude reduction when charge is shared or a multiple-site event is
recorded. The measured energy resolution was 3% FWHM at 662 keV, a good result for
a large-diameter chamber.
Finally, a pixellated anode has been proposed for use in HPXe ionization
chambers by Feng et al. [65]. In this concept, four 1-cm x 1-cm pixels are metallized
onto an insulating plate and form independent anode channels; they are separated by a
noncollecting grid which is held at a lower potential to help focus the electrons onto the
pixels. Each pixel is electrostatically shielded from charges moving far from the anode
plane by the neighboring pixels and noncollecting grid. This concept introduces more
readout complexity due to the potential for many independent anodes, but is resistant to
microphonic degradation. In addition, electronic noise broadening is expected to be quite
small due to the relatively small detector capacitance for each pixel. Finally, this concept
has the potential for three-dimensional position sensing of gamma-ray interaction
positions, which would be a new contribution to HPXe ionization chambers.
1.4.5 Signal Noise Filtering
Seifert et al. took a different approach to the problem of microphonic degradation
of energy spectra [66]. Instead of developing robust hardware to functionally replace the
Frisch grid, signal noise filtering was applied using a standard gridded HPXe detector.
Pulse waveforms were collected using LabVIEW and converted to frequency domain;
16
this allowed for investigation of the problematic frequencies arising from microphonics.
Using digital filtering to remove each problematic frequency band individually, the
detector’s energy spectrum could be restored to its non-perturbed resolution when the
system was exposed to only that particular driving frequency. It is unclear whether this
method could be implemented in real time, or if it could be used when the detector is
exposed to a broad range of driving frequencies. Nonetheless, the fact that this technique
could effectively filter out the microphonic noise from a gridded detector is a noteworthy
contribution. The principle advantage of this approach is that it can be used with
standard hardware, so in this respect it is a very practical technique. The downside of this
approach is the need for substantial computations to perform the noise filtering, which is
unattractive because although a common detector can be used, it must now be attached to
specialized data processing equipment, which will reduce operational flexibility.
1.5 Sensing the Gamma-Ray Interaction Position in HPXe
The gamma ray’s interaction position can be useful information to collect. The
interaction location might allow one to determine the direction of the incoming gamma
rays; perform detector diagnostics, such as ensuring proper biasing and gas purity by
looking at the distribution of interaction positions; or improve spectroscopic results by
correcting pulse amplitudes based upon interaction positions, or possibly by rejecting
events that register in undesired locations. Undesired locations might include volumes
inside the detector where the electric field is known to be nonuniform, or possibly
locations physically located outside the detector, which would indicate an improperly-
recorded event.
There have been very few attempts at position sensing in HPXe ionization
chambers; most have been in the last decade. Generally these methods fall into one of
two broad categories: methods employing scintillation collection to convert drift time to
interaction position, and methods using the ratio of signals on two or more electrodes to
deduce the interaction coordinate. Many of the detectors mentioned in the previous
section (Section 1.4) are capable in principle of some type of position sensing; however,
only those experiments in which position sensing capabilities were explicitly reported are
summarized here.
17
Figure 1.7. Position sensing along the X coordinate (as shown). Scintillation light is generated in the varying gap between electrodes I and II, then directed by a light guide (10) to a window (7) for collection by a photomultiplier tube (8).
1.5.1 Position Sensing with Scintillation Time Stamps
The first reported attempt at position sensing in HPXe was by Goleminov,
Rodionov, and Chepel [67]. This detector utilized a single-drift-region planar HPXe
chamber; the noteworthy design feature was that the spacing between the Frisch grid and
anode was not uniform, but changed linearly as a function of lateral location. A
photomultiplier tube was coupled to the detector, and by using an appropriate electric
field between the Frisch grid and anode plane, scintillation light could be produced
during an electron’s full traversal of the gap. Thus, the length of the scintillation light
pulse was linearly related to the position of the gamma ray interaction. A schematic is
shown in Figure 1.7.
Tepper and Losee [56] reported a design incorporating a photomultiplier tube
mated to a dual-drift-region planar HPXe chamber; only the initial scintillation light was
measured in this system, with the end of the pulse determined by the anode signal
formation. The authors pointed out that xenon scintillation light decays with two distinct
decay times, a fast component at 2.2 ns and a slow component at 27 ns, and that heavy
ion tracks have an enhanced fast component due to the higher ionization density. Thus,
the scintillation pulse can be used to reject background events originating from cosmic
18
rays, in addition to converting total drift time to distance for calculating the interaction
depth between the anode and cathode. It should be noted that no experimental results
were reported from this system.
1.5.2 Position Sensing with Electrode Signal Ratios
Athanasiades, Lacy, and Sun conceived a cylindrical gridless HPXe chamber that
utilizes a segmented cathode [68]. The central anode provides an estimation of the
charge deposited in the detector, and the ratio between the charge induced on each of the
six cathode strips and the anode can be used to localize the radial and azimuthal
interaction coordinates. The calculated radial coordinate can then be used to correct the
measured anode signal amplitude for the radial dependence of the induced charge. No
experimental results were presented, but it is likely that problems might arise from two
sources: (i) the authors assumed electronic noise from each channel would be quite low,
but it seem likely that there will be a large capacitance between cathode strips that
substantially increases electronic noise in those signal channels; and (ii) using the
methodology presented it is difficult to know how multiple-site events would be
registered.
A similar concept was employed in an experiment by Athanasiades et al. [69], but
this detector used a single insulating cathode covered with a slightly conductive paint.
Six pickup wires were placed just outside the chamber in 60° intervals. The overall
position-sensing methodology remained the same as previously described. The position
resolution was less than 2 mm FWHM in the radial direction, and between 5.6° and 7.2°
FWHM in the azimuthal direction. This particular system likely performs better than the
segmented cathode due to the small capacitance between pickup wires.
1.6 Objectives of This Work
The goal of this study is to develop a coplanar-anode HPXe chamber that
provides competitive energy resolution while being insusceptible to microphonic
degradation. It is felt that coplanar anodes, while being limited more severely by
electronic noise than other configurations, provide a rigid structure that is easy to operate
and has the potential for excellent signal amplitude uniformity throughout the entire
19
active volume. Coplanar anodes provide the ability to perform position sensing in a very
straightforward manner, which will be demonstrated; this information is useful for device
diagnostics and spectroscopic improvements, and position sensing can be performed with
a coplanar-anode chamber more easily than any of the aforementioned position-sensing
techniques. Finally, an essential point of this effort is to understand the contributions to
photopeak broadening and to determine if these degradation sources can be suppressed.
This will allow an understanding of whether coplanar anodes can provide an attractive
alternative to gridded HPXe ionization chambers in the future.
The presentation of the aforementioned work in this dissertation will begin with a
theoretical discussion of charge induction on detector electrodes, and discuss how
coplanar anodes functionally replace the Frisch grid, as well as the premise for position
sensing using coplanar anodes. Detector modeling will follow, with a description of the
different simulations performed to accurately model detector response. Established
theory and modeling results will then be applied to the detector design; the basic concept
is discussed, as is electrode optimization and the critical biasing conditions for complete
electron collection upon the desired electrode.
Detector filling will be discussed alongside the initial testing, with results from
the first gamma-ray detection experiments. Position-sensing theory specific to this
detector geometry is developed next, and its application to the experiment is presented
alongside detailed simulations. Finally, the factors impacting photopeak energy
resolution are investigated through a series of simulations and experiments to
quantitatively determine the contribution of each source, and whether improvements can
be made. The conclusions will follow, along with suggestions for future work.
20
CHAPTER 2
SIGNAL INDUCTION THEORY
2.1 Signal Induction on a Conductor by Moving Point Charges
Radiation interactions in ionization chambers create electron and positive ion
distributions at the interaction site. To register an interaction, an electric field is applied
across the detection volume, causing the electrons and ions to drift toward the anode and
cathode, respectively. As these point charges move through the detector, they induce
measurable charge on the electrodes. It is important to understand the dependence of this
signal upon the instantaneous position of the point charges over the entire duration of
charge movement: this information can then be used to extract the desired information of
the initial radiation interaction.
Figure 2.1. An illustration of important parameters in the Shockley-Ramo Theorem.
The instantaneous induced current on an electrode from the motion of a point
charge is given by the Shockley-Ramo Theorem [70, 71]. This theorem is derived using
q
11w S
ϕ = 2
0w Sϕ =
( )i t
v
21
a conservation of energy argument by He [72], so a detailed derivation will not be
presented at this time. Referring to Figure 2.1, the induced current ( )i t on an electrode
due to a moving point charge q is proportional to the scalar product of the local particle
velocity v and a parameter called the weighting field, denoted wE :
( )i t q= ⋅ wv E (2.1)
The weighting field and a related parameter, the weighting potential wϕ , are
governed by the equations and boundary conditions
1
2
2 0
1
0
w
w
w S
w S
ϕϕ
ϕ
ϕ
∇ =
−∇ =
=
=
wE (2.2)
In (2.2) the boundary conditions are defined along two surface sets: surface 1S
refers to the conducting surface—or surfaces, if multiple conductors form an electrode—
for which the induced charge is of interest, while surface 2S encompasses all other
conducting surfaces. Equations (2.2) show that the weighting field and the weighting
potential are calculated using the same equations as those governing the operating electric
field and potential distributions in electrostatics, with the exceptions of the boundary
conditions and the exclusion of space charge distributions (even if space charge is
present, it does not affect the weighting potential distribution). The weighting potential
wϕ is dimensionless; the weighting field wE has units of inverse distance.
Physically, the weighting potential is the normalized instantaneous charge ( )Q x
induced on surface 1S by point charge q located at position x . This can be easily seen
after manipulating equation (2.1) into a form that is often more convenient for analysis,
equation (2.4). Let us first convert the scalar product of the particle velocity and
weighting field into an equivalent expression:
22
,k w kk
k w
k k
w
v E
dx ddt dx
ddt
ϕ
ϕ
⋅ =
⎛ ⎞= −⎜ ⎟
⎝ ⎠
= −
∑
∑
wv E
(2.3)
Since the induced current is the time derivative of the induced charge ( )Q x , the final
expression for the induced charge is simply equation (2.3) substituted into (2.1) and
integrated over the drift time of the moving charged particle. This expression is
( ) ( ) ( )0w wQ q ϕ ϕ⎡ ⎤Δ = − −⎣ ⎦x x x (2.4)
It is important to point out that although charge induction on electrodes is
determined by the weighting potential and is therefore purely geometrical in nature,
charge motion through the detector is governed by the operating potential distribution and
is therefore influenced not only by geometry, but also by the actual biases applied to the
electrodes.
2.2 Single-Polarity Charge Sensing
Single-polarity charge sensing refers to signal generation that is only sensitive to
the motion of either positive or negative charges (but not both). Often it is beneficial to
design electrodes that are sensitive only to electron motion. Let us examine signals
generated in a simple two-electrode detector as the motivation for single-polarity charge
sensing. Then the Frisch grid and coplanar anodes will be introduced as methods of
implementing single-polarity charge sensing.
2.2.1 Planar Two-Electrode Systems
Consider a gamma-ray interaction in a detector that creates n electron-ion pairs in
a simple detector with a single planar anode located at 1x = and a cathode forming the
plane 0x = . The drifting electrons move toward the anode with total charge eq ne= − ,
23
where e is the elementary charge; the ions drift toward the cathode with a cumulative
charge of iq ne= + . Using the Shockley-Ramo Theorem in the form of equation (2.4),
the total charge induced on the anode is the sum of the electron and ion contributions:
( ) ( )( ) ( ) ( )( ) ( )
( )( ) ( ) ( )( ) ( )
( )( ) ( )( )
0 0
0 0
anode e w e w i w i w
w e w w i w
w e w i
Q t q t q t
ne x t x ne x t x
ne x t x t
ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ
ϕ ϕ
⎡ ⎤ ⎡ ⎤Δ = − − − −⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤= − − −⎣ ⎦ ⎣ ⎦⎡ ⎤= −⎣ ⎦
x x x x
(2.5)
The weighting potential is calculated by setting the anode surface to unity and the
cathode to zero. The solution is very simple in this geometry: w xϕ = .
If the electrons and positive ions are fully collected at the anode and cathode,
respectively, then ( )( ) 1w ex tϕ → ∞ = and ( )( ) 0w ix tϕ → ∞ = in equation (2.5). In this
case, then the final anode signal becomes anodeQ neΔ = ; since the final signal is
proportional to the number of initial ionizations, it becomes straightforward to make
gamma-ray spectrometry measurements.
Often, though, full collection of the positive ions is impractical or even
impossible. In gaseous detection media such as HPXe, positive ions have very small drift
velocities, on the order of 1/1000th that of electrons. It becomes quite impractical to wait
for complete ion collection, as this limits one to low count-rate applications and imposes
difficult requirements upon the pulse-height measurement electronics. Therefore, it is
desirable to measure the pulse only over a short duration, often just longer than the
maximum electron collection time. In this case, however, the ion movement is
negligible, and the induced charge on the anode is ( )01anodeQ ne xΔ ≅ − . The final signal
is now not only a function of the number of ionizations created in the radiation
interaction, but also the interaction location; spectroscopy now is much more difficult, if
possible at all.
24
2.2.2 The Frisch Grid
The Frisch grid [73] was developed to restore the proportionality between the
anode signal and the amount of charge created in ionization chambers. Frisch grids are
located near the anode (at location 1x P= − for this example) and are typically formed
from several small-diameter wires; many of these wires are stretched to form a parallel
array or a crossed grid. The effect of this grid is to create a nearly-ideal anode weighting
potential distribution. The grid is operated at a potential that maintains a sufficient field
at all points in the detector for full charge collection, and such that electrons can pass
through the grid without premature collection.
To calculate the weighting potential, the anode is again set to unity, the cathode
once more to zero; the Frisch grid is set to zero, since we are interested in the anode
signal, and the Frisch grid is an independent electrode. The weighting potential
distribution is now
( )[ )
( ) [ ]
0 0,1
1, 1 ,1
w
x Px x P
x PP
ϕ⎧ ∈ −⎪= ⎨ − −
∈ −⎪⎩
(2.6)
A schematic of the gridded ionization chamber and the weighting potential distribution is
shown in Figure 2.2.
Returning to the first line of equation (2.5), the anode signal is generally a
summation of the electron and positive ion contributions. With a Frisch grid
appropriately spaced from the anode, however, for nearly all interactions the point of
origin will be between the Frisch grid and the cathode. For these events, the positive ions
are always located in a region where the weighting potential is zero: it does not matter if
the ion drift velocity allows for significant motion during the signal readout time or not.
Thus, the Frisch grid is one configuration that allows for single-polarity charge sensing,
since only the motion of electrons will generate an anode signal. Furthermore, as long as
the electrons are created between the cathode and the grid, ( )0 0w xϕ = , the final anode
signal is anodeQ neΔ = , and proportionality to the number of ionizations is restored.
25
Figure 2.2. A schematic of a gridded ionization chamber (top). The weighting potential distribution in a gridded chamber (bottom).
2.2.3 Coplanar Grid Anodes
The coplanar grid was first reported by Luke in 1994 as a method of reproducing
the effect of the Frisch grid in CdZnTe room-temperature semiconductor detectors [74].
CdZnTe suffers from a poor hole mobility-lifetime product, which creates the same
position dependence as the slow drift of positive ions in HPXe gas. Due to the immense
challenges of placing a Frisch grid electrode inside the semiconductor, an alternative was
necessary.
Coplanar grid anodes are formed from a symmetric pattern of strips that are
connected into two independent anode sets. See Figure 2.3 for a schematic of a detector
utilizing coplanar anodes. As charges drift in the detector, they induce a signal on each
anode as governed by the Shockley-Ramo Theorem. Figure 2.3 also shows the weighting
potential of each electrode. The signal of either anode is obviously not independent of
position. Yet, by designing the two anodes such that the induced charge is uniformly
26
distributed between the anodes throughout much of the volume, the weighting potential
difference is independent of position, except for within a small region near the coplanar
anodes. Electrons created through most of the detection volume will create an anode
difference signal equal to their full charge if properly collected, and hole motion will not
be sensed because holes will be moving through the bulk region. Thus, coplanar grid
anodes can reproduce the single-polarity charge sensing functionality of a Frisch grid
when signal subtraction is implemented.
Figure 2.3. A coplanar grid anode with cathode 1, collecting anode 2, and noncollecting anode 3 (top). The anode weighting potential difference is largely independent of position (bottom).
Let us assume that N electrons are created by a gamma ray interacting in the
detector. The signals on electrodes 2 and 3 (as shown in the figure) are
27
( )
( )
2 2 21
3 3 31
Nj jf i
j
Nj jf i
j
Q e
Q e
ϕ ϕ
ϕ ϕ
=
=
Δ = − −
Δ = − −
∑
∑ (2.7)
In these equations, the subscripts i and f denote the weighting potential at the initial and
final electron positions, and the superscript j refers to a specific particle. Due to the
weighting potential uniformity, for nearly all interaction locations it is true that 2 3j ji iϕ ϕ=
for every electron. The measured signal difference will then be
( )
( )2
2 3
2 2 3 31 1
2 31
ji
diff
N Nj j j jf i f i
j i
Nj jf f
j
Q Q Q
e e
e
ϕ
ϕ ϕ ϕ ϕ
ϕ ϕ
= =
=
Δ = Δ − Δ
⎛ ⎞⎜ ⎟= − − + −⎜ ⎟⎝ ⎠
= − −
∑ ∑
∑
(2.8)
Equation (2.8) shows that diffQ NeΔ = − if every electron is collected at electrode
2, in which case 2 1jfϕ = and 3 0j
fϕ = for each electron. Thus, every electron carries a
weight of +1. However, if some electrons are collected at electrode 3, then for these
particles 2 0jfϕ = and 3 1j
fϕ = , giving them a weight of -1. This will reduce the measured
signal amplitude, which is undesirable. To counter this problem, a potential difference is
applied between electrodes 2 and 3 that directs all electrons to electrode 2 for proper
collection. Due to this biasing, electrode 2 is denoted the collecting anode, whereas
electrode 3 is called the noncollecting anode. Thus, for coplanar grid anodes to be
successfully employed, three conditions must be met: the two anodes must have uniform
weighting potential through most of the volume; the measured signal must be the
difference of the two anodes; and a bias must be applied between the anode sets to ensure
proper electron collection.
The proper bias between the two anodes can be calculated theoretically using
electrostatics. To fully collect all electrons at the collecting anode, there must be no field
28
lines inside the detector that originate at the noncollecting anode, since electron drift
opposes the field line orientation. This condition can be ensured by finding the critical
interanode bias that satisfies the condition
1
0zz
ϕ
=
∂≤
∂ (2.9)
under the center of a noncollecting anode strip, with the coordinate system used in Figure
2.3 [75]. In this equation ϕ refers to the operating potential distribution, not the
weighting potential. The center of a noncollecting strip is the critical location due to the
periodic nature of the potential distribution in planes that are parallel to the anode
surface: if electrons created directly underneath the strip’s center drift to the collecting
anode, then electrons created at any other location will as well. Biasing the detector in
accordance with equation (2.9) creates a saddle point directly underneath the
noncollecting anode strip; electrons moving toward the noncollecting anode will reach a
point at which they cannot continue toward that strip due to a falling potential, so they
will drift upward in potential toward one of the neighboring collecting anode strips.
Equation (2.9) assumes that electrons exactly follow the field lines established by
the applied electrode biases. This is generally a good assumption, as drifting electrons
collide frequently with atoms in the detection medium, which causes the electron energy
distribution to be nearly in thermal equilibrium with the atoms in the detection medium.
A more exact approach is to change the critical condition slightly such that the potential
drop between the saddle point and the noncollecting anode surface is at least as large as
the kinetic energy of a drifting electron [76]. However, since electrons in common
detectors are essentially in thermal equilibrium with the surrounding medium, their
kinetic energies are well below 1 eV, and equation (2.9) is a justifiable approximation.
2.2.4 Position Sensing Using Coplanar Grid Anodes
Position sensing capabilities provide useful information for detector
characterization and spectroscopic performance improvement. For example, the ability to
observe spectral changes as a function of interaction location allows one to determine if
29
the detector has nonuniform material properties in the case of a semiconductor, or
insufficiently-pure gas in the case of HPXe ionization chambers. Likewise, since
electron trapping in a CdZnTe crystal or varying recombination in HPXe tend to shift the
photopeak location as a function of interaction location, this effect can be compensated if
the interaction coordinates can be calculated.
After first reporting the development of coplanar grid anodes, Paul Luke observed
that although the difference of the anode signals is independent of interaction location,
the amplitudes of the individual anode signals are a function of the interaction depth
between the cathode and anodes, and that a ratio of these signals could provide
information on the interaction depth [77]. He et al. pointed out that the cathode
weighting potential in planar devices is a linear function of depth multiplied by the
amount of drifting charge in the detector, whereas the individual anode signals can have
slight variations for a given depth; thus, the ratio of the cathode signal to the subtracted
anode signal gives a better approximation of the interaction depth [78]. Denoting the
cathode and anode difference signals as catS and subS , respectively,
( ) ( )catcat
sub sub
k E f dS f dS k E
γ
γ
⋅ ⋅≅ ∝
⋅ (2.10)
where Eγ is the energy deposited by the gamma ray, catk and subk constants of
proportionality between the deposited energy and the respective cathode and subtracted
anode signals, d the interaction depth, and ( )f d a function of the depth, which is linear
for a planar geometry. The measured depth should vary between 0 and 1, with near-
anode events resulting in a depth near 0 and cathode-side events a depth near 1. The
constants catk and subk should theoretically be equal if the same readout electronics are
used for the cathode and subtracted anode signals; however, different equipment settings,
such as different shaping time constants, may become necessary and destroy this equality.
In addition, ballistic deficit may noticeably reduce the cathode signal gain relative to the
anode signal, again resulting in unequal gain.
30
This depth-sensing technique could be applied to a gridded ion chamber, although
the depth calculation would not be quite as accurate because the cathode signal varies
linearly between the cathode and Frisch grid, but is zero between the grid and the anode.
Thus, all interactions between the grid and the anode would give 0d = , and events in the
rest of the detector would have to be scaled for the slight offset between the calculated
and true depth.
It is interesting to note that equation (2.10) is valid only when the positive charge
carriers are completely immobile; although this is not strictly true in practice, the better
the approximation, the more accurate the calculated depth. The reason is that when the
positive charge carriers move, they induce signals on the two anodes and the cathode;
however, for most events in a coplanar-anode detector, the anode weighting potentials
will be nearly identical along the entire path of the positive charge carriers, producing an
anode difference signal that is insensitive to positive carrier motion. Therefore, the
calculated depth for these events will be artificially large, since the cathode signal
amplitude will increase with positive charge carrier motion, whereas the anode difference
will generally be insensitive to those charges.
In this simple planar geometry, the positive carriers will induce a signal
contribution that is uniform throughout most of the device, which will simply introduce a
constant offset between the calculated and true depth. However, very near the cathode
the positive carriers will reach the cathode and the cathode signal will equal Eγ in this
region, leading to a calculated depth of 1d = for all events where the positive carriers
can be fully collected. As the positive carrier drift distance increases, so will the fraction
of the detector where these charges are fully collected at the cathode, and the depth
calculation method will become invalid. Events generated very near the anodes are
unique in that the positive carriers can induce unequal charge on the coplanar anodes,
albeit for a very short path length. This will reduce the measured depth offset by
increasing the denominator as well as the numerator in equation (2.10).
Loss of charge to recombination or trapping should be proportional to the number
of drifting charge carriers for all events at a given depth, assuming uniform material
properties and only a depth variation in the electric field. As the depth is varied, this
constant of proportionality can also change due to the position-dependence of charge
31
loss. This means that if the measured gamma-ray depositions are tallied in depth bins
that are sufficiently fine, the location of gamma lines from one depth to the next will shift
slightly due to this changing charge loss fraction. To correct this undesirable result, a
constant gain can be applied to each depth bin that aligns common photopeak centroids.
This gain will vary from one channel to the next, but will be valid from measurement to
measurement as long as the state of the detector is not changed (i.e., different biasing
conditions, a new fill gas mixture, or a different readout electronics configuration). The
power of this technique is demonstrated in Figure 2.4, which shows both the spectrum as
a function of depth and the effect of depth correction on the overall spectrum in a
coplanar grid CdZnTe detector [79].
Figure 2.4. A 137Cs depth-separated energy spectrum in coplanar-grid CdZnTe (top). Applying depth correction improves the energy resolution from 5.62% to 2.06% FWHM (bottom).
32
CHAPTER 3
DETECTOR DESIGN
3.1 Design Concept
The objective of this body of work is to develop a coplanar-anode HPXe chamber
that provides competitive energy resolution (3.5 to 4.0% FWHM at 662 keV for large-
diameter detectors) while being insusceptible to microphonic degradation of measured
energy spectra. Previous coplanar-anode HPXe designs employed: (i) a helical anode in
a cylindrical geometry and (ii) a spiral anode in a planar geometry [60, 61]. As discussed
in Section 1.4.4, the helical anode structure suffered from incomplete charge collection
problems that degraded the measured energy resolution, as electrons became trapped on
the anode support column prior to arrival at the collecting anode. The spiral anode
structure avoided the charge collection problems, but the planar geometry has two
distinct disadvantages: the detector capacitance is generally larger in this geometry than
in a cylindrical design, which increases electronic noise; and a relatively small portion of
the total gas volume is active, whereas the cylindrical geometry can have an active gas
volume of nearly 100%.
Thus, a desirable detector design would use the cylindrical geometry for improved
electronic noise and detection efficiency, but an alternative coplanar anode design must
be developed that does not rely upon the helical anode and its charge collection problems.
An interesting model for this alternative design is the dual-anode cylindrical ionization
chamber (DACIC) concept, also presented in Section 1.4.4, which features two
symmetric anode wires stretched axially in a cylindrical HPXe ionization chamber [58,
62]. Due to the azimuthal weighting potential asymmetry arising from just two anode
33
wires coupled with the charge sharing problems stemming from both anodes being held
at the same potential, the DACIC device is limited in spectroscopic performance.
Let us consider a geometry not unlike the DACIC device as the basis for the
proposed coplanar anode HPXe design. Instead of just two thin wires, multiple anode
wires will be used to improve the azimuthal weighting potential symmetry; to create the
symmetry, the wires will be oriented axially along the surface of an imaginary cylinder
with equal spacing between wires. The wires will be of relatively large diameter, making
them less sensitive to microphonic vibration. Finally, the wires will be connected into
two independent anodes and held at different operating biases, thus defining collecting
and noncollecting anodes, and signal subtraction will be employed for proper coplanar
operation.
The existing HPXe cylindrical chambers with the helical anodes will be retrofitted
with the new anode design: the pressure vessel dimensions are therefore constrained. The
variable design parameters are the number of wires (although the number of anodes is
always two, the number of wires per anode can be changed); the diameter of the
individual wires; and the radius at which the wires are centered with respect to the
detection volume’s central axis.
The dimensions of the pressure vessel and all internal components are considered
invariant in this analysis, since they have been defined by earlier designs. The pressure
vessel and its inserts have an outer diameter of 4.25 inches (108 mm), and a total wall
thickness of 0.125 inch (3.18 mm). This means the cathode radius is set at 2 inches (50.8
mm). The pressure vessel is 8 inches long (203.2 mm); the length of the detection
volume is 4 inches (101.6 mm). For design purposes, a density of 0.5 g/cm3 was
assumed, as this is the gas density used in previous detectors. The dielectric constant of
this gas is approximated as 1.12rε = [80], and the conductivity by 0σ = .
The three design parameters that can be varied in this analysis are the thickness of
the anode wires, their location relative to the center of the detector, and the total number
of wires used. Let us first consider the wire diameter. Since the intent of this project is
to remove susceptibility to microphonics, a sturdy wire is required; the tradeoff is that the
detector capacitance (and thus the associated electronic noise) increases dramatically with
increasing wire diameter. Although much analysis could be performed on the optimal
34
wire thickness, a wire diameter of 1 mm has been chosen without supporting analysis.
However, at this point a proof-of-principle design is of interest to determine whether the
anode configuration has potential to perform competitively as a spectrometer. Should
this be the case, further analysis can be performed to truly optimize the chamber’s
coupled spectroscopic and vibration properties. Similarly, the wire offset from the
detector’s central axis is also a free parameter, but in this design it is held constant at 12.7
mm. This offset is used because it is known from helical-anode detector experiments to
provide an acceptable electric field throughout the detector’s sensitive volume.
Table 3.1. A summary of fixed design parameters.
Parameter Design Value
Detection volume length 101.6 mm
Cathode diameter 101.6 mm
Anode displacement from central axis 12.7 mm
Anode wire diameter 1 mm
Xenon gas density 0.5 g/cm3
Figure 3.1. Cross-sectional schematics of the proposed detector showing end (left) and side (right) views. Anode wires are black, Macor insulators are gray, and white areas designate the gas volumes.
35
The anode wires are centered at a radius of 12.7 mm relative to the central axis of
the chamber. The wires are spaced at regular intervals along the surface of the imaginary
cylinder defined by this radius: for example, a design utilizing 8 wires has a separation of
45° between adjacent wires. The active wire length is 101.6 mm, which is defined by the
pressure vessel and its internal components; however, the true wire length necessary for
capacitance calculations is probably closer to 140 mm, as the wires pass through holes
drilled in 12.7 mm-thick Macor insulating plates on either end of the detection volume.
Table 3.1 summarizes the important design parameters. Figure 3.1 shows schematics of
the proposed design for a 12-wire anode geometry.
3.2 Electrostatic Potential Theory
It is desirable to determine the operating potential throughout the detector as a
function of the applied electrode biases, as well as the weighting potential distribution in
the detector. This information can be very useful in designing, operating, and
understanding the detector. For example, the weighting potential allows one to determine
how much charge will be induced on the anodes for each event, which can be used to
calculate expected pulse-height spectra, thus permitting analysis of different detector
configurations. The operating potential, once determined, can be used to determine how
the charges formed in an event actually move through the chamber, what biases need to
be applied to create an electric field sufficient to collect all charges, and what the charge
collection time is.
3.2.1 General Potential Theory
Let us first attempt to derive the solution for generic boundary conditions, which
can be replaced later with actual values for determining the weighting potential or
operating potential distributions. To solve for the general potential as a function of
position inside the detector, ( ), ,r zϕ θ , one solves Poisson’s equation (3.1):
( )2 , ,r zρ θϕ
ε∇ = − (3.1)
36
In this equation cylindrical coordinates are being used, so r refers to the radius,
θ the azimuthal angle, and z the axial position. The geometry is defined such that one
of the collecting anode wires is centered at 0θ = and the origin of the coordinate system
lies on the central axis of the detector, with the exact center of the detector corresponding
to the plane 0z = . The symbol ε is the permittivity of the xenon gas, and ρ the
distributed space charge.
There are three boundary conditions in this system, corresponding to the
potentials applied at the cathode surface, catϕ ; the collecting anode wires, CAϕ ; and the
noncollecting anode wires, NCAϕ . Let the solution process begins with the following
assumptions:
1. axial variations in ( ), ,r zϕ θ are negligible,
2. the solution to the differential equation is separable in the radial and azimuthal
coordinates → ( ) ( ) ( ),r R rϕ θ θ= ⋅Θ ,
3. the space charge is negligible at all locations → 0ρ = , and
4. the anode wires are two dimensional, having zero thickness in the radial direction.
Assumption (4) is necessary due to a limitation in the separation of variables method
which requires all boundaries of the solution region to be coordinate surfaces, where one
variable remains constant [81]. Using the aforementioned assumptions, the governing
differential equation (3.1) can now be transformed from
( )2 2
2 2 2
=0 0
, ,1 1 -r z
rr r r r z
ρ θϕ ϕ ϕθ ε
=
∂ ∂ ∂ ∂⎛ ⎞ + + =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠
to
2
2
λλ
1 0r d dR drR dr dr dθ
−+
Θ⎛ ⎞ + =⎜ ⎟ Θ⎝ ⎠ , (3.2)
37
where the parameter λ is the separation constant. The polarity of λ was chosen for
convenience.
Let us first turn our attention to the differential equation describing solution
behavior in the azimuthal direction,
2
2 0dd
λθΘ
+ Θ = (3.3)
To be rigorous, the separation constant cannot immediately be assumed positive; all
possible values must be considered. It is fairly simple to show that when λ is negative
only the trivial solution is possible. For this case, one arrives at the solution
( ) 1 2c e c eλ θ λ θθ − − −Θ = +
Since the solution must be periodic due to the symmetry within the detector, the only way
to satisfy the regularity of the solution is for 1 2 0c c= = . Let us now examine the case
0λ = . Considering equation (3.3) again, the solution is
( ) 3 4c cθ θΘ = +
To satisfy the periodicity requirements, the constant c4 must be zero. Finally, let us
consider the solution of (3.3) when λ is positive. This solution can satisfy the
periodicity requirements without forcing constants of integration to zero:
( ) ( ) ( )5 6cos sinc cθ λ θ λ θΘ = +
At this point the constant λ can be determined. The argument to the cosine and
sine terms must be 2π-periodic when the angle separating two adjacent collecting anode
38
wires is entered. If there are a total of wN wires (both collecting and noncollecting), then
the corresponding angle is 4 wNπ , and the eigenvalues λ are
, 0,1, 2, ...2
wNn nλ = = (3.4)
Now let us consider the differential equation describing the potential variations
with radius,
r d dRrR dr dr
λ⎛ ⎞ =⎜ ⎟⎝ ⎠
(3.5)
This equation should be solved for the cases in which nontrivial solutions to equation
(3.3) were obtained, so that a general solution may be obtained for Poisson’s equation.
When 0λ = , the solution to equation (3.5) is easily verified to be
( ) ( )7 8lnR r c r c= + ;
in this result, both constants can be nonzero and still satisfy basic solution properties, so
they must be determined when the boundary conditions are applied. For λ positive, the
general solution to equation (3.5) is given by the dimensionless equation
( ) 9 10cat cat
r rR r c cR R
λ λ+ −⎛ ⎞ ⎛ ⎞
= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
where catR denotes the cathode radius. The constants 9c and 10c will not be determined
until later, when the boundary conditions are applied to the overall solution.
Since the differential equation (3.2) is linear and homogeneous, the principle of
superposition is used to combine the results developed for nontrivial cases. The general
solution for the electrostatic potential is equation (3.6):
39
( )
( )
( )
0 0
1
, ln
cos
sin
n n
n n
cat
n n ncat cat
n
n n ncat cat
rr A BR
r rA BR R
r rC DR R
λ λ
λ λ
ϕ θ
λ θ
λ θ
+ −
∞
+ −=
⎛ ⎞= + ⎜ ⎟
⎝ ⎠⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞⎪ ⎪⎢ ⎥+⎜ ⎟ ⎜ ⎟⎪ ⎪⎢ ⎥⎝ ⎠ ⎝ ⎠⎪⎣ ⎦ ⎪+ ⎨ ⎬
⎡ ⎤⎪ ⎪⎛ ⎞ ⎛ ⎞⎢ ⎥+ +⎪ ⎪⎜ ⎟ ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠ ⎝ ⎠⎣ ⎦⎩ ⎭
∑ (3.6)
Now the boundary conditions will be applied. First it is noted that, due to the
symmetry of the geometry and the imposed boundary conditions, the electrostatic
potential solution is a symmetric function with a period defined by the number of anode
wires in the detector. Also, the coordinate system has been defined such that the
potential solution is symmetric about 0θ = , and is therefore an even function. It is a fact
that if a function is even, then its Fourier series only contains cosine terms [82]. Thus in
equation (3.6), 0n nC D= = for all n . The complete solution is now
( ) ( )0 01
, ln cosn n
n n nncat cat cat
r r rr A B A BR R R
λ λ
ϕ θ λ θ+ −
∞
=
⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥= + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦∑ (3.7)
At this point, it is necessary to treat the detector in two parts: the regions with
radii (i) greater than and (ii) less than anR , the radius of the imaginary cylinder on which
the anode wires are placed. The requirement of two separate solution regions is apparent
if one tries to use just one region for the entire detection volume. If first the boundary
condition at the cathode surface is applied to equation (3.7) and then the solution is
examined as 0r → , the only finite solution is ( ), catrϕ θ ϕ= , which is not true if the
anodes are held at unique potentials.
For the outer region an catR r R≤ ≤ , one should first examine the boundary
condition at the cathode surface. It is an easy matter to show that the only general
solution to this boundary condition (i.e., for arbitrary values of θ ) requires
40
1. 0out
catA ϕ= , and
2. 0out outn nA B+ = .
For the region 0 anr R≤ ≤ , one applies the appropriate boundary condition at the
center of the detector in lieu of that at the cathode surface. This boundary condition
places the constraint 0 0in innB B= = in equation (3.7); otherwise the corresponding terms
would be unbounded near the detector’s center, which is unphysical. The general
equations describing the potential in the inner and outer regions are
( ) ( )
( ) ( )
01
01
, cos
, ln cos
n
n n
in in inn n
n cat
out out outcat n n
ncat cat cat
rr A AR
r r rr B AR R R
λ
λ λ
ϕ θ λ θ
ϕ θ ϕ λ θ
+∞
=
+ −∞
=
⎛ ⎞= + ⎜ ⎟
⎝ ⎠⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥= + + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦
∑
∑ (3.8)
The inner and outer solution regions meet at an interface located at anr R= . The
boundary condition at this interface is now considered, as it will allow us to determine
0inA , in
nA , outnA , and 0
outB explicitly. Since the distribution of the potential along this
imaginary surface is unknown at this time, let us generally refer to the interface potential
as ( )intϕ θ . The coefficients An and B0 are determined via the Euler formula [83], which
invokes orthogonality arguments to pick out one coefficient at a time:
41
( )
( ) ( )
( ) ( )
( )
2
0 int2
2
int2
2
int2
2
0 int2
4
cos2
cos
2
4
4 ln
w
w
w
w
w
w
w
w
Nin w
N
Nin w catn n
an N
Nout wn n
Nan an
cat cat
Nout w
catwNan
cat
NA d
N RA dR
NA dR RR R
NB dNR
R
π
π
λ π
π
π
λ λπ
π
π
ϕ θ θπ
ϕ θ λ θ θπ
ϕ θ λ θ θ
π
πϕ θ θ ϕπ
−
−
−−
−
=
⎛ ⎞= ⎜ ⎟
⎝ ⎠
=⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥−⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
⎛ ⎞= −⎜⎜⎛ ⎞ ⎝ ⎠⎜ ⎟
⎝ ⎠
∫
∫
∫
∫ ⎟⎟
(3.9)
It is evident that the functional form of ( )intϕ θ is necessary to solve the Euler
formula. This is not known a priori, as the anode wires occupy only a fraction of the
circumference of the cylinder defined by anR . At this point, it is necessary to use
numerical methods to determine the Fourier constants. Let us proceed with the
information in hand for further analysis.
3.2.2 Weighting Potential Representation
The weighting potential in cylindrical coplanar-anode radiation detectors was
investigated previously by He and Khachaturian [84]; the theoretical framework used by
those authors is implemented in this chapter. The technique of coplanar anodes requires
the subtraction of induced signals; the Shockley-Ramo theorem [72] can be used to show
that the difference between the collecting anode’s weighting potential at the point of
charge carrier creation and that of the noncollecting anode is a parameter of interest for
optimizing signal amplitude uniformity. Let us return to equation (2.8), assigning
collecting anode (CA) status to electrode 2 and making electrode 3 the noncollecting
anode (NCA), and assume all electrons are ideally collected at the collecting anode:
42
( ) ( )
( )
( ),
, , , ,1 1
, , , ,1 1 0
,1
1
jdiff i
diff CA NCA
N Nj j j j
CA f CA i NCA f NCA ij i
Nj j j j
CA f NCA f CA i NCA ij
Nj
diff ij
Q Q Q
e e
e
e
ϕ
ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ
ϕ
= =
=
=
Δ = Δ − Δ
= − − + −
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟= − − − −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
= − −
∑ ∑
∑
∑
(3.10)
In this case, the final measured signal is a summation over all N electrons of the
anode weighting potential difference at the point of charge creation, ,j
diff iϕ . Ideally ,j
diff iϕ
is zero everywhere, which makes the measured signal amplitude a function of only the
number of electrons created in the radiation interaction.
Since Laplace’s equation (3.1) is linear and homogeneous, it is trivial to show that
because the weighting potential distributions of both the collecting and noncollecting
anodes obey Laplace’s equation, then the difference of the two will obey the same
equation:
2 2 0CA NCAϕ ϕ∇ = ∇ = (3.11)
( )2 2 2 2 0CA NCA CA NCA diffϕ ϕ ϕ ϕ ϕ∇ − ∇ = ∇ − = ∇ = (3.12)
Table 3.2. Boundary conditions for the various weighting potentials.
Conducting Surface CAϕ Value NCAϕ Value diffϕ Value
Cathode 0 0 0
Collecting Anode 1 0 1
Noncollecting Anode 0 1 -1
Now that the differential equation is being solved directly for the weighting
potential difference, a consideration of boundary conditions is necessary. Table 3.2 gives
43
the boundary conditions for the collecting and noncollecting anode weighting potentials;
subtraction yields the boundary conditions for the weighting potential difference diffϕ .
3.3 Chamber Optimization
To optimize the number of wires in the anode structure, let us assume that the
detector is biased optimally such that all electrons created in an interaction drift to the
collecting anode. In this idealized case, the uniformity of the anode weighting potential
difference diffϕ will be of utmost importance. Let us now return to the problem of
approximating the weighting potential via a Fourier series. Electronic noise contributions
will be considered later, and the optimal combination of charge induction and electronic
noise will determine the optimal geometry.
It is worth stating at this point that in the following sections functional
approximations are made to simulated data without rigorous error analysis. This is not
expected to have a significant impact upon the optimization study, as the modeling
employs many idealizations that are expected to introduce significant uncertainly.
3.3.1 Numerically Determining the Weighting Potential Fourier Coefficients
There are three difficulties in solving equations (3.9) for the Fourier coefficients:
the function ( )intϕ θ is not known a priori; except for a few special cases of ( )intϕ θ , the
integrals cannot be solved without the help of mathematical software; and finally, there
are an infinite number of terms in the Fourier series. Fortunately, these complications
can be overcome using the Maxwell 3D [85] and Mathematica [86] software packages.
Maxwell 3D is a finite-element software package for solving electrostatic
problems. By creating the geometry as accurately as possible and using the proper
material properties, the potential distribution can be calculated throughout the geometry.
This allows the estimation of ( )intϕ θ by plotting the numerical solution along the arc
anr R= . Mathematica can be used for solving the difficult integrals in (3.9) and for
determining how many terms in the Fourier series are needed for a close approximation
of the true solution.
44
Let us investigate the convergence and accuracy of coupling equations (3.8) and
(3.9) with numerical simulations modeling the proposed geometry: anode wires of radius
0.5 mmwR = centered at radius 12.7 mmanR = relative to the central axis of the
chamber, and cathode radius 50.8 mmcatR = . Consider three different cases: 2, 4, and 12
anode wires. To determine the Fourier coefficients, approximate ( )intdiffϕ θ with the
piecewise-continuous function
( ) ( )
0
0 0
int 0 0
0 0
0
2 21 ; ,
2cot ; ,2
, 1 ; ,
2cot ; ,2
2 21 ; ,
w w
w
w
diffan
w
w
w w
N N
NN
R
NN
N N
π πθ θ
πα θ θ θ θ
ϕ θ θ θ θ
πα θ θ θ θ
π πθ θ
⎧ ⎛ ⎞− ∈ − − +⎪ ⎜ ⎟
⎝ ⎠⎪⎪⎪ ⎛ ⎞⎛ ⎞− ∈ − + −⎜ ⎟⎪ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎪⎪⎪= + ∈ − +⎨⎪⎪ ⎛ ⎞⎛ ⎞+ ∈ + −⎪ ⎜ ⎟⎜ ⎟
⎝ ⎠⎪ ⎝ ⎠⎪⎪ ⎛ ⎞
− ∈ −⎪ ⎜ ⎟⎪ ⎝ ⎠⎩
(3.13)
In (3.13), the symbol 0θ is a configuration-dependent constant that represents the
angle at which the cotangent term value equals one:
( )10
2 tanwN
θ α−=
Figure 3.2 plots simulation data from the Maxwell 3D code against distributions
calculated with equation (3.13) for a fixed radius of 12.7 mm, which is the location of the
interface between the inner and outer solution regions. The differences between the
simulation data and the approximating function are large only in small regions near the
edge of the anode wires, and seem to decrease as the number of anode wires increases.
45
The coefficient α used to calculate the weighting potentials estimated by equation (3.13)
was adjusted to the best fit by eye, and is listed for each case in Table 3.3.
Table 3.3. Values of α used to calculate the weighting potential difference in Figure 3.2.
Number of Anode Wires α
2 0.20
4 0.25
12 0.45
Figure 3.2. Comparison of one period of the simulated weighting potential difference and the approximation along the solution interface for 2, 4, and 12 anode wires.
Now that an approximation for the interface weighting potential difference intdiffϕ
has been established, let us focus on calculating the Fourier series in (3.8). Terms where
the summation index n is even have Fourier coefficients of zero, since these indices
46
result in cosine terms with local maxima at locations where the true function being
approximated is zero. Thus, to compute only the nonzero Fourier series terms, a simple
modification of nλ in equations (3.8) can be performed:
( ), 0,1, 2 2 1 , 0,1, 22 2
w wn n
N Nn n n nλ λ= = → = + =… …
Figure 3.3. Comparison of the Maxwell 3D simulated anode weighting potential difference and the first 25 terms of the Fourier series for 2, 4, and 12 wires.
Now it is appropriate to examine the weighting potential difference calculated
with equations (3.8) and compare them to the Maxwell 3D simulation results. Figure 3.3
is a comparison of the first 25 nonzero terms of each Fourier series to the Maxwell-
simulated weighting potential difference. The abscissa represents the radial coordinate
for each data point; the data presented is along the plane 0θ = , directly through the
center of a collecting anode wire. The overestimation of the subtracted weighting
potential in the vicinity of the anode wires is likely due to the imperfect fit of the
47
assumed boundary-condition profiles in Figure 3.2, although the treatment of the wires as
2-dimensional surfaces should also contribute to the deviation. Notice in Figure 3.2 that
the distribution of intdiffϕ calculated for 2 anode wires is not as accurate as that for 12
wires, thus translating into a poorer estimation of the weighting potential in Figure 3.3.
The results are satisfactory when there are more than 2 anode wires, especially
when considering that interactions will occur more frequently near the cathode due to the
cylindrical geometry, and the equations are quite accurate in this region. Also, the
weighting potential distribution for fixed radius is maximized at 0θ = ; as the azimuthal
angle is shifted toward an adjacent noncollecting anode wire, the difference between the
estimated and true weighting potential will generally improve.
It is important to investigate whether summing over 25 Fourier cosine terms will
indeed result in a converged answer. To examine this assumption, Figure 3.4 shows the
convergence of the solutions when the first 1, 5, 10, and 25 nonzero terms are included.
From the figures, it appears that approximately 10 terms is sufficient for a converged
solution.
It has now been shown that 10-term expansions of equations (3.8) and (3.9)
provide an acceptable approximation to the true distribution along one particular radial
segment. The next step is to investigate whether these equations correctly model the
behavior with changing azimuthal angle. To test this, Maxwell 3D-simulated data at a
fixed radius of 20 mm has been generated over an entire angular period (e.g., from the
center of one noncollecting anode to the center of the next noncollecting anode). A
radius of 20 mm has been chosen because the weighting potential slowly becomes more
sinusoidal in shape as the radius increases, so near the cathode a very smooth cosine
shape should exist, which is not a challenging test scenario. Near the anode, though,
some non-sinusoidal behavior can be observed due to the proximity of the anode wires,
and this is expected to be more difficult to model.
48
(a) (b)
(c)
Figure 3.4. Demonstrating the convergence of the Fourier series for (a) 2 anode wires, (b) 4 wires, and (c) 12 wires. The 10 term expansion cannot be seen as it lies directly underneath the 25 term expansion.
A comparison between these simulations and the results from equation (3.8) can
be found in Figure 3.5 below. In this figure, the series expansion is scaled by a factor
corresponding to the difference in theory and simulation demonstrated in Figure 3.3. The
predicted solution behavior is in acceptable agreement with the true distribution. In fact,
the differences are only noticeable for 2 anode wires, and this is again because of the
49
poor approximation of boundary conditions at the anode radius (see Figure 3.2). Figure
3.5 also demonstrates convergence of the Fourier approximation as the number of
summed series terms is varied again from 1 to 25 terms. Series convergence is
acceptable with very few terms, as changes cannot be noticed beyond 5 terms at this
particular radius. Only the convergence of the 2-wire simulation is shown, since
convergence is even more rapid when the number of anode wires is increased.
(a) (b)
Figure 3.5. (a) Comparing the simulated and calculated ( )20 mm,diffϕ θ . The calculated distribution is scaled by the amplitude difference found in Figure 3.3 to isolate the azimuthal changes. (b) Convergence of the Fourier series expansion for 2 anode wires.
It is appropriate at this point to test the validity of the first assumption made for
the mathematical treatment of the general potential solution, that the anode wires are
sufficiently long to neglect axial variations in the potential. This investigation can easily
be made by examining the Maxwell 3D results for various geometries along lines of
constant radius and azimuth. The first plot, Figure 3.6, shows the simulated anode
weighting potential difference along the fixed line ( ) ( ), 20 mm,0r θ = ; the number of
simulated anode wires is varied from 2 to 16. This fixed line lies in the plane intersecting
a collecting anode and the central axis of the detector, and the line is not far away from
the anode wire. In this coordinate system the origin lies at the geometrical center of the
detector, so the plot shows the variations from the detector midpoint out to the end of the
50
cylindrical volume; due to symmetry, the entire length need not be considered. The
anode weighting potential difference is very uniform as a function of axial position
throughout the detection volume, 0 50.8 mmz≤ ≤ , especially when there are at least 4
anode wires.
It is possible that this uniformity is misleading due to the location’s proximity to
an anode, which fixes the weighting potential to a uniform value of unity at the wire.
Therefore, Figure 3.7 shows a similar plot when the number of anode wires is fixed at 8,
the azimuthal angle at 10º, and the radius is varied. Again, the uniformity throughout the
detection volume is fairly good. Thus, it seems that the assumption of axial uniformity is
reasonable.
Figure 3.6. Axial variations in the anode weighting potential difference for 2 to 16 anode wires along the line ( ) ( ), 20 mm,0r θ = .
51
Figure 3.7. Axial variations in the anode weighting potential difference for 8 anode wires along lines with 10θ = ° .
3.3.2 Charge Induction Uniformity Dependence Upon the Number of Wires
Now that both an acceptable mathematical formulation and the more accurate
numerical simulations of the weighting potential inside the proposed chamber have been
developed, the next step in detector optimization is to study the effect of the number of
anode wires on the uniformity of charge induction throughout the entire working volume.
Referring to equation (3.10), it is apparent that for optimal charge induction in which
diffQ NeΔ = for interactions at any point in the detection volume, it is necessary for the
anode weighting potential difference to satisfy 0diffϕ = everywhere in the xenon gas.
At this point, let us use the Maxwell 3D electrostatic solver for the analysis due to
its improved accuracy. Using equation (3.12) and the boundary conditions supplied in
Table 3.2 it is possible to solve for the anode weighting potential difference directly. The
52
simulations use the geometry and materials described in Section 3.1 with the number of
anode wires varying from 2 to 16. Cylinders created in Maxwell 3D are approximated by
a user-defined number of rectangular segments; it was found via trial and error that 40
segments is a sufficient number. The results of these simulations along a line intersecting
a collecting anode wire on the plane 0z = are shown in Figure 3.8. This particular line
was chosen because diffϕ will be least ideal in the proximity of an anode wire, so the plot
shows the worst case (near the noncollecting anode diffϕ has the same magnitude but
opposite sign).
Figure 3.8. The dependence of ( ), 0, 0diff r zϕ θ = = on the number of anode wires.
From a single-polarity charge sensing standpoint, it is obvious that increasing the
number of wires results in more ideal charge induction. At some point adding more wires
will have severely diminished returns, since the improvements in charge induction will
occur mainly for interactions in a vanishingly small volume near the anode wires. There
53
are two practical limitations on the number of wires, one of which is that at some point it
will become difficult to construct the detector in such a way that it does not arc when the
electrodes are biased. The other limit is reached sooner, and it is related to electronic
noise.
3.3.3 Electronic Noise as a Function of the Number of Anode Wires
It has now been established that charge induction uniformity favors an anode
structure with more wires. Let us consider the effect of wire number on electronic noise.
The preamplifiers that will be used in experiments will be Amptek A250s with 2SK152
JFETs; Figure 3.9, reprinted from the A250 data sheet, provides an estimation of
electronic noise as a function of detector capacitance for a shaping time of 2 μs, shown as
data series 4 (in blue) [87]. A 2 μs shaping time is likely shorter than what will be used
in the experiments due to the slow drift of electrons through HPXe chambers, but since
this is the only data available giving electronic noise as a function of detector
capacitance, let us use it anyway. To quantify the electronic noise, it is necessary to first
calculate the detector capacitance, dC , which can easily be computed using Maxwell 3D
simulations.
Figure 3.9. The expected A250 noise as a function of detector capacitance and FET.
54
To determine the detector capacitance, first note the relationships between the
charge on an electrode cq , the capacitance dC between the signal electrode and its
neighboring conductors, the potential difference ϕΔ between conducting surfaces, and
the displacement field D in the bulk medium:
c dq C ϕ= Δ (3.14)
cS
q d= ⋅∫ D a (3.15)
S is the surface of the signal electrode; in the case of the collecting or noncollecting
anode, this is the total surface of all wires comprising the electrode.
Let us set the potential on the signal electrode to 1ϕ = , and let 0ϕ = on all other
conducting surfaces. Now 1ϕΔ = and simple algebra leads to expression (3.16) for the
capacitance, which is a formula that can be calculated directly in Maxwell 3D:
dS
C d= ⋅∫ D a (3.16)
Using the geometries previously created in Maxwell 3D for weighting potential
calculations, the detector capacitance was calculated for anode structures up to 16 wires.
Combining this information with the data presented in Figure 3.9, the preamplifier noise
can be estimated using the concept of equivalent noise charge (ENC). The ENC is the
amount of charge that, if suddenly placed on the signal electrode, would create an output
signal equal in magnitude to the standard deviation of the peak broadening caused purely
by electronic noise; ENC is often presented in units of electrons [8]. The ENC can be
converted to FWHM with knowledge of w , the mean energy to create an ionization. The
symbol e− denotes an electron.
( ) ( ) ( )eV 2.35 eVFWHM ENC e w e− −= ⋅ ⋅ (3.17)
55
The results obtained using equations (3.16), (3.17), and Figure 3.9 are presented
in Table 3.4. The mean energy to create an ionization used is 21.9eVw = [8].
Table 3.4. Amptek A250 electronic noise contribution as a function of anode geometry.
Number of Wires dC , pF ENC, electrons FWHM, keV
4 3.09 141 7.26
8 6.64 146 7.52
10 8.34 151 7.78
12 10.78 156 8.03
14 13.08 161 8.29
16 16.10 166 8.55
The electronic noise from the preamplifier favors fewer wires in the anode
geometry; however, the weighting potential data becomes more ideal when the number of
anode wires is increased. Thus a final decision on the optimal geometry is best
determined by performing simulations including both effects, which can combine them
into one response function. Because the calculated width due to electronic noise
broadening changes very little from one case to the next, it is expected that the optimal
geometry will have a relatively large number of wires, since weighting potential
improvements are fairly significant at low anode wire numbers.
3.3.4 The Optimal Balance of Electronic Noise and Weighting Potential
Geant4 simulations are used to determine the optimal number of anode wires.
Geant4 [88], distributed by the CERN laboratory, is a Monte Carlo code that tracks the
history initiated by each incident particle; one of the attractive attributes of Geant4 for
this application is that tallies are easily constructed by the user to give nearly any
information desired over a simulation run, whereas other widely-used codes such as
MCNP5, distributed by Los Alamos National Laboratory, are very limited in output. In
addition, it is relatively easy to insert code that performs certain calculations at each
interaction location using the simulation data; this allows one to easily implement
important concepts such as weighting potential and Fano statistics. The code requires the
56
incident radiation source to be defined, as well as the geometry, materials, physical
processes of interest, and output tallies.
The simulations to select the optimal anode structure include the effects of Fano
statistics, anode weighting potential difference, and preamplifier electronic noise. The
anode weighting potential difference distribution used is the 10-term Fourier series
approximation represented by equations (3.8), (3.9), and (3.13); it accounts for radial and
azimuthal variations in the measured signal amplitude. The electronic noise is sampled
from a normal distribution characterized by the ENC values of Table 3.4. To determine
the total electronic noise for the system, the ENC for one A250 is multiplied by 2 to
simplistically account for the use of two preamplifiers in the experiments. The Fano
statistics sample from a normal distribution characterized by mean value depE w and
standard deviation depF E w , where depE and F are the energy deposited in an
interaction and the Fano factor, respectively.
Assumptions made for these Geant4 simulations are:
1. axial effects are negligible;
2. it is sufficient to track only energy transfers from the primary gamma ray;
3. the charge cloud created in an interaction is negligibly small;
4. the detector is biased sufficiently to collect all electrons at the collecting anode;
5. positive ions remain motionless for the duration of signal formation;
6. ballistic deficit is negligible even for a 2 μs shaping time;
7. there is no charge recombination;
8. there is no charge diffusion; and
9. the Fourier series approximations introduce negligible error.
Of these assumptions, numbers 1, 3, 4, 5, 7, and 8 are not expected to cause significant
deviations from the experiment. Assumption 2 is expected to enhance the photopeak
somewhat due to the inclusion of energy carried out of the sensitive volume by x-rays or
electrons diffusing into the wall, energy which in reality should be excluded from
calculations. This assumption will also result in the omission of x-ray and escape peaks
from the simulated spectra, which is acceptable for this particular study. Assumption 6
57
really is two separate suppositions: that a shaping time of 2 μs is acceptable
experimentally, and that no amplitude deficit will be measured for any event.
Assumptions 6 and 9 are the most questionable, but will be analyzed in Chapter 6 for
validity.
Simulated spectra are generated for a source of 137Cs gamma rays flooding one
end of the pressure vessel. The modeled detector geometry includes all steel, Macor
insulator, and HPXe spaces; minor volumes such as the anode wires and connectors are
not modeled. Since the only physical processes that contribute to the energy spectra in
these simulations are Compton scattering and photoelectric absorption of the initial
gamma rays, spectral features such as x-ray escape peaks are absent. Figure 3.10 shows
the spectral variations as a function of the number of anode wires, along with a close-up
of the photopeak region.
The result of this analysis is that 12 anode wires (2 groups of 6 wires each) is
expected to be the optimal configuration for best energy resolution: fewer wires will
broaden the photopeak due to larger deviations of the weighting potential from the ideal
distribution, while using more than 12 wires degrades resolution from the associated
increase in electronic noise. The predicted energy resolution of 2.3% FWHM at 662 keV
is quite good; it is not only better than other HPXe detectors of similar dimension, but is
comparable to small-diameter chambers. The spectral feature around 200 keV is the
traditional backscatter peak, which comes from the inclusion of the steel pressure vessel
and the Macor shell in the Geant4 simulation geometry. There are a significant number
of counts between the Compton edge and the photopeak; these are due to multiple
Compton scatters in the working gas. The presence of counts at energies greater than the
gamma ray’s actual energy, also noted in [89], is an artifact of the signal subtraction step.
Essentially it is possible for gamma-ray interactions to occur in locations where the net
induced charge on the collecting anode is close to the charge generated in the interaction,
while the noncollecting anode’s induced charge is negative. The subtraction step yields a
net signal appearing to be greater than the gamma ray’s actual energy, producing a
continuum that extends up to double the gamma ray’s energy. If Figure 3.8 had shown
diffϕ for azimuthal angles near a noncollecting anode wire, negative values of diffϕ would
58
be shown. Inserting these values into equation (3.10) gives artificially-high energy
measurements.
Figure 3.10. Simulated 137Cs energy spectra for different anode wire numbers (top). Zooming in on the peak region shows 12 wires to give the best resolution, 2.3% FWHM (bottom).
59
3.3.5 The Importance of Signal Subtraction
To demonstrate the significance of signal subtraction in this geometry, let us
compare the previous results to simulations of a spectrum formed from the collecting
anode’s preamplifier signal only. In this case, the weighting potential difference of the
two anodes is replaced by just the collecting anode weighting potential, which is much
less ideal – see Figure 3.11. The weighting potential analysis was again simulated using
Maxwell 3D. It is interesting to compare Figure 3.8 and Figure 3.11 to note that without
signal subtraction, the collecting anode’s weighting potential distribution for 12 wires is
less ideal than even the 2 wire case when signal subtraction is employed. This effect is
somewhat offset by the improved electronic noise, since the subtracted signal’s noise
term is the quadrature sum of the noise from both anodes’ preamplifiers.
Figure 3.11. The simulated anode weighting potential difference compared to the weighting potential of the collecting anode only.
To quantitatively compare the expected pulse height spectra measured from the
collecting anode and the subtraction circuit outputs, Geant4 simulations are repeated for
60
the 12 wire case with the only differences being (i) the electronic noise term and (ii) the
weighting potential for induced charge calculations. Figure 3.12 displays the results of
these two simulations for a 137Cs gamma-ray source irradiating the detector uniformly
over one end of the pressure vessel. Note that using only the collecting anode signal is
expected to give much worse energy spectra – there is no defined photopeak. This
coplanar-anode HPXe detector can therefore be expected to provide a viable alternative
to gridded HPXe chambers only when signal differencing is employed.
Figure 3.12. A comparison of simulated 137Cs energy spectra for a 12-wire anode when signal subtraction is employed versus when only the collecting anode signal is utilized.
3.3.6 Estimating the Critical Electrode Biasing of the HPXe Chamber
Now that a geometry has been chosen, it is possible to use equations (3.8) and
(3.9) in conjunction with the discussion of Section 2.2.3 to determine the critical biases
61
for full electron collection. For full collection of electrons at the collecting anode, the
condition
2, 0sw
r Rr Nϕ πθ
⎛ ⎞∂= = ≥⎜ ⎟∂ ⎝ ⎠
(3.18)
must be met, where sR denotes the outer surface coordinate of the noncollecting anode
wire. Let us also assume that a minimum electric field magnitude minE must be
maintained near the cathode for optimal extraction of electrons from the ionization site.
It is easy to show from equation (3.8) that the azimuthal component of the electric field is
zero at the cathode surface; thus, a second restriction on the operating bias is
( ) min,catr R Erϕ θ∂
− = ≥∂
(3.19)
To solve for the critical conditions, let us determine the collecting anode and
cathode operating biases CAϕ and catϕ for which the inequalities (3.18) and (3.19) are just
satisfied (the noncollecting anode is assumed to be grounded for this calculation). Only
the region between the anodes and cathode is of concern: the region inside the anodes is
known in advance to suffer from weak fields, but can be neglected due to its small
fraction of the total gas space. For equations (3.9) to be solved, it is necessary to develop
an expression describing the potential ( )intϕ θ along the interface between the inner and
outer solution regions. For simplicity, let us assume a simple cosine function oscillating
between 0 and CAϕ as the azimuthal angle shifts from the noncollecting to the collecting
anode:
( )int 1 cos2 2CA wNϕϕ θ θ
⎡ ⎤⎛ ⎞= + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ (3.20)
62
In this simplified case, the Fourier coefficients outnA of equation (3.9) are greatly
simplified, as only the first term in each summation is nonzero:
1 2 2
0
2
12
ln
w w
out CAN N
an an
cat cat
out CAcat
an
cat
AR RR R
BRR
ϕ
ϕ ϕ
−=
⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥−⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
⎛ ⎞= −⎜ ⎟⎛ ⎞ ⎝ ⎠⎜ ⎟⎝ ⎠
(3.21)
The coefficients in (3.21) can be substituted into equation (3.8) to obtain an
approximation for the operating potential distribution throughout the detector, which can
then be inserted into equations (3.18) and (3.19), thus producing a system of linear
equations which can be solved easily using simple algebra:
( ) ( ) ( ) ( )
( ) ( )( ) ( )( ) ( )
min2 2
2 1 2 1
2 2
1 1 1ln 2 ln
1 1 1 0ln 2 ln 2
w w
w w
w w
wcat CA catN N
an cat an cat an cat an cat
N Ns cat s catw s
cat CAN Nan cat an cat cat an cat an cat
N E RR R R R R R R R
R R R RN RR R R R R R R R R
ϕ ϕ
ϕ ϕ
−
− − −
−
⎡ ⎤− + − = −⎢ ⎥
−⎢ ⎥⎣ ⎦⎡ ⎤+
− + − =⎢ ⎥−⎢ ⎥⎣ ⎦
(3.22)
To obtain numerical values for the critical biases, one must substitute the known
design values of 12.7 mmanR = , 50.8mmcatR = , and 12wN = . The minimum electric
field strength is taken as the value at which the drift velocity of electrons saturates in the
HPXe gas, which for the design density of 0.5 g/cm3 is approximately min 1250V cmE =
[48]. The final unknown is the location of the noncollecting anode surface, sR . In the
initial assumptions used to obtain equations (3.22), the wire has zero thickness in the
radial direction; thus, it seems that s anR R= . This may lead to inaccuracies due to the
true wire surface being located at a different position than assumed in the model; in
addition, there are other assumptions that may introduce significant error into the
63
calculation, such as the assumption of a pure cosine distribution for ( )intϕ θ , which
distorts the modeled field in the vicinity of the noncollecting anode wire. Nevertheless,
the following approximate critical biases can be used as a guide for initial testing of the
HPXe chamber, and can later be assessed for accuracy:
7.75kV
2.12 kV
critcatcrit
CA
V
V
= −
= + (3.23)
64
CHAPTER 4
INITIAL EXPERIMENTS
4.1 Detector Preparation and Gas Filling
In Chapter 3 a coplanar-anode HPXe ionization chamber design was developed
and optimized. In this section the detector construction will be described and shown, as
will the gas purification and filling system used.
4.1.1 Detector Construction
Two detectors have been assembled using the design outlined in Section 3.1 with
a 12-wire anode geometry. The pressure vessels are made of stainless steel: the outer
wall is 1 mm thick to allow gamma rays to penetrate with minimal absorption and
scattering, although the endplates are much thicker, 0.75 inch, for structural integrity.
The three vessel components (two endplates and one central wall) are welded to form a
strong and leak-proof union. The volume inside the pressure vessel is 6.5 inches in
length with diameter 4.25 inches.
The active volume is supported by a smaller Macor cylinder placed inside the
pressure vessel. Macor is chosen because it is an excellent insulator—its bulk resistivity
is greater than 1016 Ω·cm [90]—which is important for minimizing leakage current
between the electrodes and also for isolating the high-voltage cathode from the grounded
pressure vessel. In addition, Macor does not outgas when it is properly baked [90],
meaning it will not contaminate the purified HPXe gas. The cathode is formed by a layer
of silver that is metallized on the inner wall of this Macor cylinder. The Macor also
provides support for the anode structure, as the BeCu wires are fed through small holes
machined in the Macor and fastened on either end. The primary detection volume,
65
bounded by this Macor structure, has both length and diameter 101.6 mm. The Macor is
secured inside the pressure vessel by Macor spacers that hold the internal components in
place.
Figure 4.1. A photograph of the anode structure in a test assembly. The connection scheme used to group the wires into two signal electrodes is apparent.
Figure 4.2. A photograph of the detector prior to final assembly.
66
Figure 4.3. A photograph of the Macor spacer (center, white) that separates the pressure vessel (top) from the main Macor internals (bottom).
Figure 4.1 shows the anode wires in a test assembly; this picture illustrates the
arrangement of the wires and how connections are made at the end of the detector to
cluster the wires into two anodes (collecting and noncollecting groups of six wires each).
Figure 4.2 shows the anode wires protruding from a Macor structural shell prior to
insertion into the steel pressure vessel. Figure 4.3 portrays one end spacer used to secure
the internal components.
4.1.2 Gas Purification and Detector Filling
Extreme care must be taken to properly purify the xenon fill gas prior to filling.
The presence of electronegative molecules severely degrades chamber performance; to
ensure good performance, impurities such as O2, CO, CO2, and organic molecules must
be reduced in concentration to at most 0.5 ppm [49, 91]. To obtain this high gas purity,
research-grade xenon is introduced into a spark chamber with titanium electrodes. A
continuous discharge between these electrodes is established, which creates a cloud of
titanium dust. This dust absorbs the electronegative impurities very efficiently; one
benefit of this method is that the dust can continue to further purify the gas even if the
67
power supply to the electrodes is shut off. This purification technique can produce
electron lifetimes in the HPXe gas that rise above 1 ms, ensuring that at most only a few
percent of electrons are lost prematurely due to attachment.
In addition to gas purification, the interior detector surfaces and the inner walls of
the transfer pipes between the spark chamber and the detector must be scrubbed of all
impurities that might contaminate the gas. To accomplish this goal, the appropriate
components were wrapped in heat tape and held at a temperature of about 125ºC for
several days to accelerate outgassing of impurities from the surfaces. The system was
placed under a high vacuum (10-7 torr) to evacuate the system of all gas and to ultimately
remove impurities that diffuse from the piping and detector surfaces. A schematic of a
typical purification and filling station is shown in Figure 4.4, reprinted from Bolotnikov
and Ramsey [91]. The getter, oxisorb, and sampling cylinder can be used for initial
purification of the xenon gas, but in this case the gas was introduced directly into the
spark purifier via the test chamber port prior to detector baking.
Figure 4.4. A typical HPXe purification and filling station using a spark purifier.
Once the baking and purification processes were complete, the heat tape was
removed, the valve lineup changed, and the detector filled with HPXe gas. The final gas
density in the detectors was approximately 0.28 g/cm3 in detector HPXe1 and 0.25 g/cm3
in detector HPXe2, as determined by the measured change in detector mass and the
known volume of the gas space. These densities are well below the design value of 0.5
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g/cm3, but due to problems with the filling procedure these were the final detector states.
To estimate the corresponding critical electrode biases, the values calculated in Section
3.3.6 only need to be scaled by the experimental-to-design density ratio, since the critical
electric field for drift velocity saturation scales linearly with density.
4.2 Experiments
The initial experiments with these detectors tested some fundamental properties of
the detectors, such as detector capacitance, electronic noise, and linearity, and explored
the limits on the applied biases. These parameters are important for both characterizing
the detector and also understanding its performance.
Figure 4.5. A diagram of the capacitance measurement concept.
4.2.1 Capacitance Measurements
The capacitance between the two anodes and also between an anode and the
cathode were measured experimentally. (Due to symmetry, it was unnecessary to
measure the cathode-noncollecting anode capacitance.) A schematic of the measurement
technique is shown in Figure 4.5. The procedure used was to input a known test pulse
with amplitude inVΔ directly onto either the cathode or the noncollecting anode, and then
to measure outVΔ from the collecting anode preamplifier while this electrode is grounded.
The relationship between inVΔ and outVΔ is very simple:
inVΔ
A250
1pFfbC =
outVΔelectrodesC
QΔ
69
( )
1pF
pF
electrodes in fb out
outelectrodes
in
Q C V C V
VCV
=
Δ = Δ = Δ
Δ=
Δ
(4.1)
Thus it is straightforward to calculate the capacitance between two electrodes.
The test pulse injected on the opposing electrode is given time constants similar to
a true gamma-ray signal. The measured capacitances are found in Table 4.1; these values
are representative of both the HPXe1 and HPXe2 detectors. The total detector
capacitance is the combination of the measured values in parallel, or 21.9 ± 2.0 pF per
preamplifier. This experimental result agrees with the detector capacitance predicted
with Maxwell 3D simulations—22.8 pF.
Table 4.1. Measured capacitance values for the HPXe detectors.
Electrode Pair Cathode-Anode Anode-Anode Vessel-Anode Total Anode
C (pF) 1.6 ± 0.2 11.7 ± 1.6 8.6 ± 1.2 21.9 ± 2.0
4.2.2 Electronic Noise Measurements
Electronic noise is important to quantify because it places a lower limit on a
detector’s energy resolution. It is also possible to compare the actual electronic noise of a
system, which is normally dominated by the preamplifier, against the manufacturer’s data
to establish whether or not the equipment is performing as expected. To make this type
of comparison, the electronic noise measurement must be calibrated on an absolute scale.
The measurement begins with a separate Amptek A250 preamplifier connected to
each anode of the HPXe chamber. A test pulse of known amplitude inVΔ is injected into
one of the preamplifiers, the output of the preamplifier stage is filtered and amplified
using a shaping amplifier, and this shaped amplitude is measured using a multichannel
analyzer (MCA). Because the test pulse voltage signal inVΔ is placed across a capacitor
( )2.0 0.1 pFtestC = ± in the A250 test board, it creates a charge at the operational
amplifier’s inverting terminal of magnitude
70
test test inQ C VΔ = Δ (4.2)
By using multiple known test pulse amplitudes, the MCA can be calibrated in units of
charge. Assuming the test pulse amplitude has negligible variation, the electronic noise
of the system in units of equivalent noise charge (ENC) is simply the standard deviation
of the measured pulse height distribution, or assuming a normal distribution of pulse
heights,
( ) ( )19
12.35 1.602 10
FWHM C eENC eC
−−
−=×
(4.3)
This calibration can be performed using multiple shaping time constants to
determine the optimal time constant for minimizing the noise contribution. The
electronic noise contribution of the preamplifier is expected to pass through a minimum
at some shaping time corresponding to a balance between series noise—such as thermal
noise in the FET and feedback resistor that is dominant at short shaping times—and
parallel noise, an example being the detector leakage current shot noise that becomes
problematic at long shaping times [8]. It is important to note that the optimal shaping
time constant does not necessarily correspond to the best measured energy resolution: if
the charge transit time through the detector is not short compared to the shaping time,
ballistic deficit concerns will increase the ideal shaping time constant.
Using the measurement procedure described above, the electronic noise of the
system was quantified at various shaping times. For these experiments the detector was
not biased; the results may change somewhat under operating conditions if, for example,
leakage current between the electrodes becomes significant. The output of the
subtraction circuit was analyzed since this is the primary signal of interest; the measured
system noise is expected to be larger than the output of just a single preamplifier because
the difference signal incorporates noise from both preamplifiers as well as any
fluctuations introduced by the subtraction circuit. Two different shaping amplifiers were
used to test a larger sample of shaping time constants. The ORTEC 672 shaping
amplifier proved to be lower noise than the Canberra 243, which can be verified easily by
71
comparing the results from these two amplifiers at a shaping time of 2 μs. All ENC
results are displayed in Figure 4.6, with the corresponding limit on energy resolution for
HPXe shown in the bottom panel. The optimal electronic noise is achieved with a
shaping time of 6 μs, placing a lower limit on energy resolution of 2.7% FWHM at 662
keV.
Figure 4.6. Measured ENC (top) and the corresponding energy resolution limit (bottom) as a function of shaping time for the HPXe detection system.
72
4.2.3 Preamplifier Waveforms
As the detector operating biases were increased during the experiments, the
preamplifier waveforms were monitored on an oscilloscope. As the cathode bias was
changed from 0 to -4500 V, the measured rise time of the subtraction circuit output
shortened noticeably, from around 50 μs to approximately 5 μs, as presented in Table 4.2.
It is important to remember that although the actual charge transit time is much longer
than these values, the implementation of coplanar anodes and signal subtraction creates
an idealized signal that registers no response from charge carrier motion through most of
the detector, followed by a sharp rise when the electrons have approached the anode
structure. This difference can clearly be seen in Figure 4.7, which shows the measured
collecting and noncollecting anode preamplifier waveforms along with the subtraction
circuit output. The cathode bias is -4000 V for this particular measurement; the
collecting anode is held at +1300 V. The individual anodes show a total charge transit
time of near 40 μs, while the 5 μs subtracted output rise time allows a shaping time of 16
μs to be used.
Figure 4.7. Measured collecting (green) and noncollecting (blue) anode waveforms, shown with the subtraction circuit output (yellow). The horizontal axis is 100 μs long, and the vertical axis is 16 mV in amplitude.
73
Table 4.2. Subtracted signal rise time vs. cathode bias while the anodes are grounded.
Cathode Bias Subtraction Circuit Signal Rise Time
-1000 V ≥50 μs
-2000 V 25 μs
-3000 V 10 μs
-4000 V ~5 μs
4.2.4 Cathode Biasing Spectra
The purpose of the first set of measurements using a 137Cs gamma-ray source was
to determine the necessary applied cathode bias to approach complete electron collection
and minimize ballistic deficit for all events. This was achieved by recording pulse-height
spectra at 500-volt intervals and looking for the point at which spectral changes ceased.
The collecting and noncollecting anodes are both grounded for this series of experiments.
The experimental setup for this set of experiments is very simple, and is presented
in the block diagram of Figure 4.8. This figure shows one of the advantages of using
coplanar grids: other than the use of two preamplifiers and a simple subtraction circuit,
the experimental setup is quite conventional for gamma-ray spectroscopy, relying only
upon standard equipment.
Figure 4.8. A block drawing of the equipment setup for gamma-ray experiments.
HPXe Detector
CA Preamp
NCA Preamp
Cathode HVPS
CA HVPS
Subtr. Circuit
Shaping Ampl.
MCA
74
Figure 4.9 shows a subset of the data acquired from this experiment. The energy
spectra presented are unusual—there are counts in negative channel bins. This is because
the output of the subtraction circuit can be either positive or negative, depending upon the
amplitude and polarity of both the collecting and noncollecting anode signals. Since both
anodes are grounded, electrons will not drift preferentially toward one anode or the other;
furthermore, the wire symmetry makes electrons equally likely to be collected on either
anode. Thus, over a large number of interactions, the expected energy spectrum will be
symmetric about the vertical axis. To obtain this spectrum using a conventional MCA,
first a spectrum was collected with the shaping amplifier set to positive output polarity,
then a separate spectrum was collected using inverted polarity. The two halves of the
spectrum were then stitched together manually. The region around the vertical axis
appears to have no counts because of the lower-level discriminator (LLD) setting on the
multichannel analyzer.
Figure 4.9. 137Cs gamma-ray spectra as a function of the applied cathode bias when both anodes are grounded.
75
In Figure 4.9 the spectra are not quite symmetric about the vertical axis as
expected. This implies that the subtraction circuit may not be tuned perfectly, and as a
result the gain of the noncollecting anode signal is about 5% less than that of the
collecting anode. This also accounts for there being more counts per channel in the
negative portion of the spectrum—the gain difference compresses the same number of
counted events into fewer bins. The data for cathode biases beyond -4000 volts showed
no visible changes in the spectrum, implying both that nearly all of the electrons are
being collected and also that the electron drift velocity has saturated near 1 mm/μs
through most, if not all, of the detection volume.
4.2.5 Anode Biasing Spectra
Now that an appropriate cathode bias of -4000 V has been established, the next
step is to start utilizing the anodes properly by increasing the applied bias on the
collecting anode (the noncollecting anode is grounded throughout all experiments). The
goal is to shape the electric field locally around the anodes such that all electrons drift to
the collecting anode wires. As the collecting anode bias is increased, electron collection
will occur more frequently on the collecting anode than on the noncollecting anode wires,
and thus the negative subtracted signals are expected to diminish in amplitude and
frequency as the positive signal amplitudes and frequency increase.
Figure 4.10 presents a subset of the experimental data. As expected, increasing
the collecting anode bias gradually shifts the negative-polarity pulses toward positive
energies. The photopeak FWHM does not seem to improve when the collecting anode
bias is raised above 400 V, even though electron collection at the collecting anode is far
from complete at that point: this implies that other sources of peak broadening dominate
the measured FWHM. Electronic noise contributes significantly to photopeak
broadening, as quantified with a test signal. The test peak has a width of 32.3 keV, thus
placing a 4.9% FWHM lower bound on the energy resolution. The measured photopeak
resolution is 6.8% FWHM at 662 keV when employing collimators to direct the incident
radiation upon the midplane of the detector, calculated using the peak fitting routine in
the ORTEC® MAESTRO-32 MCA software [92].
76
Figure 4.10. 137Cs spectra as a function of collecting anode bias with the noncollecting anode grounded and the cathode held at -4000 V.
One easy way of experimentally estimating the critical bias for complete electron
collection at the collecting anode is to also pass the noncollecting anode’s preamplifier
signal to a shaping amplifier and register the number of counts above a fixed
discrimination level (preferably set just above the noise level). As the collecting anode
bias is increased, electrons will be collected less frequently on the noncollecting anode
wires; as a result the measured signals from this electrode will tend to become more
negative as the collecting anode bias increases. Let us consider the induced signal on the
noncollecting anode NCAQΔ using the Shockley-Ramo theorem. The total drifting charge
q is the sum of the total charges collected on the collecting and noncollecting anodes,
CAq and NCAq respectively, absent charge recombination. Then, assuming the weighting
potentials of the collecting and noncollecting anodes at the ionization site, 0CAϕ and 0
NCAϕ ,
77
are very nearly equal—as expected for a good coplanar anode design—then they can be
replaced simply by 0ϕ , and the signal under consideration is
( ) ( )0 0
0
0 1
1
CA NCANCA CA NCA
CA
Q q q
qqq
ϕ ϕ
ϕ
Δ = − − − −
⎛ ⎞= − − −⎜ ⎟
⎝ ⎠
(4.4)
Figure 4.11. A preamplifier waveform from the noncollecting anode (yellow) and the shaped signal (green). The shaping time used is 16 μs.
Optimal charge collection can be estimated as the point at which the positive
noncollecting anode counts disappear. This is demonstrated by setting CAq q= in
equation (4.4) while noting that 0 0ϕ ≥ and q is negative. Strictly this statement is not
quite true when pertaining to the filtered signal that is measured in an experiment,
because even if all electrons are properly collected, the noncollecting anode signal
transient can create a shaped signal with a significant positive lobe. This is demonstrated
by the waveforms shown in Figure 4.11: even though the overall change in preamplifier
output voltage (yellow) is negative, the signal remains positive for about 22 μs, and with
78
a 16 μs shaping time constant this generates a shaping amplifier output amplitude of +2V
(green), well above the detection threshold.
Positive noncollecting anode count data were obtained with the cathode biased at
-4500 V and a 137Cs source present; this data is presented in Figure 4.12. The lower-level
discriminator setting corresponds to about 50 keV; decreasing the discriminator setting
beyond this point introduced large system dead time fraction (greater than about 15%).
The uncertainty bars are not presented in the figure because they are too small to be
distinguished from the data points. It is evident that at 1400 V the detector is still not
optimally biased, but beyond this point discharging began and data was not collected in
the interest of protecting the equipment. The background count rate was measured, and
from this it is apparent that a large fraction of the measured counts are true source events.
Figure 4.12. The number of positive noncollecting anode counts as a function of the collecting anode bias; the source is 137Cs, and the cathode is held at -4000 V.
Monitoring the preamplifier waveforms on the oscilloscope provides a second
method of determining the critical biasing condition of the detector. If electrons are
79
drifting to the noncollecting anode, then some events will be observed to exhibit a net
positive change in the noncollecting anode’s preamplifier signal.
4.2.6 Effects of Background Radiation and Source Collimation
To determine the degree of response uniformity along the axial coordinate, the 137Cs point source was collimated using lead bricks to irradiate ¼-inch axial slices of the
detector. The collimated beam was focused upon three different locations: the center of
the detector and the gas spaces at either end of the HPXe chamber. These results were
compared to the uncollimated case in which the entire chamber was exposed to the
gamma-ray source.
Figure 4.13. The effects of background correction (red) and source collimation (blue) on the raw 137Cs spectrum (black). A test pulse is centered near channel 925.
In addition, the effect of background radiation on the measurements was
quantified by collecting a spectrum under equivalent biasing and shaping conditions
80
without the 137Cs point source present. Figure 4.13 demonstrates the effects of first
applying a background correction to the measured spectrum, then collimating the source
at the center of the detector. For this particular series of measurements, the applied
biases were -4500 V on the cathode and +1400 V on the collecting anode; the shaping
time is 16 μs, and the MCA live time is 1800 s. The net effect of these two “corrections”
is to greatly reduce the count pileup near 200 keV; about half of the total difference
comes from background events, the other half is apparently due to nonuniform response
along the chamber’s axis.
Figure 4.14. The detector response with the source collimated upon the detector center (black) and the opposing end gas spaces where the anode (red) and cathode (blue) lead wires are located.
Figure 4.14 demonstrates the effect of beam location on the detector response.
For this set of experiments the shaping time has been reduced to 12 μs; background has
been subtracted from all of the spectra presented. It is demonstrated that the end regions
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do not contribute to the photopeak at all: these volumes are evidently responsible for
much of the drastic count rate increase measured in the Compton continuum near 200
keV. The reason for this is likely due to the low fields and nonuniform weighting
potential in these end gas spaces. The cathode does not extend into these spaces, so their
boundaries are mostly formed by the pressure vessel, which is held at ground potential.
The slight differences in the spectra can be explained by the difference in field strength
on the two ends resulting from the anode lead wires penetrating one end versus the
cathode lead wire in the other end. The cathode bias wire creates a stronger field, which
results in the pulse distribution being shifted to higher energies.
Figure 4.15. The change in the photopeak region for a selection of Gaussian filters using shaping times from 8 to 16 μs.
82
4.2.7 Optimal Shaping Filter
The optimal shaping time will minimize the width of the photopeak in the
measured spectrum. This is not necessarily the same time constant as that which
minimizes electronic noise, since peak broadening due to ballistic deficit may be
important for shaping times that are not long compared to the preamplifier signal rise
time. Employing source collimation upon the central detector plane but not background
correction, 137Cs energy spectra were recorded for several shaping time constants: 8, 10,
12, and 16 μs. The applied biases were again -4500 V on the cathode and +1400 V on
the collecting anode, and the MCA live time is unchanged at 1800 s. The experimental
results are shown in Figure 4.15: clearly the 8 μs data prove that this time constant is too
small, but the other three choices give fairly close results. The 12 μs shaping time
provides slightly better energy resolution than the other time constants.
Figure 4.16. The optimized 137Cs energy spectrum exhibits a 6.0% FWHM energy resolution and greatly reduced counts in the lower part of the Compton continuum.
83
In addition, the shaping amplifier allowed either Gaussian or triangular filtering.
The triangular filter was found to be superior in similar testing, providing a measured
resolution of 6.0% vs. 6.4% FWHM at 662 keV using the MCA peak analysis software
and an identical region of interest. The final energy spectrum, which uses the optimal
filter plus collimation upon the central detector plane and a background correction, is
shown in Figure 4.16.
4.2.8 System Linearity
A final important test of the system is to determine its linearity in response to a
wide range of gamma-ray energies. Previous HPXe systems have been shown to have
excellent linearity [24, 27, 43, 50], thus a linear response is expected to be measured with
this detector.
For this experiment, the system settings were: -4500 V cathode bias and +1400 V
collecting anode bias, a triangular shaping filter with a 12 μs time constant, collimation
upon the central plane of the detector, and the MCA live time was increased to 7200 s.
Background was measured and subtracted to enhance the lower-energy peaks. The
sources used are listed in Table 4.3, along with the corresponding gamma-ray energies
and intensities [93], plus the measured photopeak centroids. The measured spectrum is
presented in Figure 4.17. The 60Co 1173.2 keV peak cannot be easily discerned from the
nearby Compton edge of the 1332.5 keV line, which lies at 1118.1 keV. The recorded
channel number as a function of gamma energy is displayed in Figure 4.18, plotted along
with the calculated linear least-squares fit line. The length of each half error bar (either
the half above or below the data point) corresponds to one standard deviation of
measured peak width, calculated from the measured FWHM and assuming a Gaussian
distribution of counts in the photopeak. It is evident from this figure that the linearity of
the HPXe detection system is quite acceptable.
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Figure 4.17. A spectrum collected with 133Ba, 137Cs, and 60Co gamma-ray sources.
Table 4.3. Sources used in the linearity measurement along with photopeak properties.
Source Energy (keV) / Intensity (%)
Measured Centroid (channels)
Measured FWHM (channels)
133Ba 81.0 keV (34.1%) 85.18 20.50 133Ba 302.9 keV (18.3%) 285.66 36.61 133Ba 356.0 keV (62.1%) 343.38 29.68 137Cs 661.7 keV (85.1%) 622.60 36.97 60Co 1173.2 keV (99.9%) - - 60Co 1332.5 keV (100.0%) 1219.38 48.25
85
Figure 4.18. The measured photopeak centroid plotted against the true gamma-ray energy (black). Total error bar lengths correspond to two measured standard deviations. The linear least-squares solution is also displayed (red).
Table 4.4. The departure of measured peak centroids from the least-squares solution.
Gamma-Ray Energy (keV)
Absolute Deviation (channels)
Relative Deviation (% of FWHM)
Relative Deviation (% of centroid)
81.0 -4.04 -19.7 -5.51
302.9 -4.60 -12.6 -1.68
356.0 5.01 16.9 1.55
661.7 7.27 19.7 1.21
1332.5 -3.70 -7.7 -0.31
The deviation of each measured photopeak centroid from the least-squares
solution is listed in Table 4.4. The maximum absolute deviation is 7.27 channels for the
86
137Cs point source, which corresponds to 8.02 keV departure from the expectation value.
The deviations are relatively small compared to the measured FWHM of each peak, with
the maximum being 19.7% of the measured FWHM. The deviations are also a small
percentage of the true gamma energies, with all cases except the lowest energy being less
than a 1.7% shift from the expected peak location.
4.2.9 Discussion of Preliminary Results
In these first experiments of the coplanar-anode HPXe chamber, the preamplifier
noise contribution was quantified as a function of amplifier shaping time. It was found
that the best noise performance occurs at a shaping time of 6 μs, which is too short to
avoid ballistic deficit when the working gas is pure xenon. A shaping time constant of
either 12 or 16 μs was used for all experiments, appropriate for the measured pulse rise
times of about 5 μs.
The detector was biased in incremental steps to determine the minimum cathode
and collecting anode voltages to ensure full and proper collecting of electrons. A cathode
bias of -4000 V was found appropriate for collection of events throughout the entire
detection volume; the minimum collecting anode bias for proper coplanar operation could
not be achieved, as discharging occurred before this operational point was reached.
The spectrum recorded with 133Ba, 137Cs, and 60Co exhibited good linearity over a
range of gamma-ray energies from 81.0 keV to 1332.5 keV. All major gamma lines
could be measured except the 1173.2 keV line from 60Co, which was obscured by the
presence of the nearby Compton edge at 1118.1 keV corresponding to the 1332.5 keV
gamma ray. The maximum deviation of a measured photopeak centroid from the least
squares solution is 8.02 keV, and all deviations fall well within the measured photopeak
FWHM.
In Figure 4.10 the collected spectra all exhibit a rather extreme change in the
Compton continuum count rate between channels 200 and 400. This effect is also present
in many published HPXe results; for example, refer to [35]. There are two common
theories used to explain this feature: one oft-cited possibility is related to the slow drift of
electrons in xenon gas, expected to be about 1 mm/μs in the high-field region near the
anodes and about 0.5 mm/μs near the cathode using published drift velocity data [94];
87
this agrees with experimentally-observed electron transit times of nearly 50 μs over the
38 mm cathode-anode spacing. In the case of gamma rays that undergo multiple
interactions inside the detector, the possibly-significant drift time difference of the
electrons compared to the shaping time constant used may cause a significant number of
the events to be counted as two small-amplitude pulses instead of being integrated into
one large-amplitude pulse: this would shift some events from high channels, possibly
including some from the photopeak, to low channels in the Compton continuum.
A second common explanation for the significant change in the Compton edge
count rate is Compton scattering within the pressure vessel, which can have a very
significant mass relative to that of the fill gas [27]. For example, in the coplanar-anode
HPXe detectors the mass of the vessel and internal structures is 4.960 kg (HPXe1) and
4.670 kg (HPXe2), whereas the mass of the added gas is 330 g for detector HPXe1 and
295 g for HPXe2. If this theory is true, the spectral feature is simply the traditional
backscatter peak observed in gamma-ray spectra, albeit with extreme prominence. This
question will be studied in more detail in Chapter 6, but it seems that the former
explanation can be discounted based upon the results of the collimation studies.
Another noteworthy property of the measured spectra is the poor energy
resolution. In the Geant4 simulations of Chapter 3 several assumptions were made that
simplified the simulations: foremost was the assumption of zero ballistic deficit even for
very short shaping times, proven false from experimental observations. As discussed
previously, the parallel noise contribution to electronic noise increases as shaping time
lengthens; this can be observed in the data of Figure 4.6. Reducing the shaping time to
the ideal of 6 μs could reduce the preamplifier noise contribution to peak broadening and
bring the measurements closer to the simulated predictions. One way of achieving this
reduction without introducing ballistic deficit problems would be to use small hydrogen
admixtures to the xenon gas, which have the effect of increasing the electron drift
velocity by a factor of up to 5 or more at moderate electric field intensities [48]. These
energy resolution matters will be studied in greater detail in Chapters 6 and 7.
88
CHAPTER 5
RADIAL POSITION SENSING
5.1 Motivation for Position Sensing
The interaction position of a gamma ray within a detector is valuable information,
if it can be measured. The interaction location can be used to apply pulse height
corrections that counteract charge recombination and weighting potential changes as a
function of position; to diagnose the detector’s operating conditions, such as ensuring
sufficient biasing and verifying HPXe gas purity; and to select interactions of interest
within the detector. As described in Section 1.5, a handful of efforts to determine
interaction coordinates in HPXe detectors have been reported in publications. These
methods have focused on either using scintillation light emitted in the ionization cloud as
a time stamp, or on using segmented perimeter electrodes—not necessarily the cathode—
and using signal ratios from each electrode to triangulate the interaction location. While
these methods work in theory, they can be challenging to employ in practice.
Scintillation light collection can be difficult due to problems mating a photomultiplier
tube to a high-pressure chamber, and the amount of light emitted by a gamma-ray
interaction in HPXe is quite minimal. The detector capacitance for the electrode strips
can be large, reducing the signal-to-noise ratio; also, it becomes necessary to process
many signals concurrently, adding a great deal of complexity to data acquisition and
processing.
The incorporation of coplanar anodes into HPXe ionization chambers permits a
simpler position sensing method. The two anode signals can be used to give not only the
position-independent signal amplitude, but also the approximate radial coordinate of the
gamma-ray interaction. The method used to determine the radial position of the gamma-
89
ray interaction is analogous to the depth sensing technique in coplanar-anode CdZnTe
detectors described in Section 2.2.4. The following sections will first introduce the
theoretical framework for calculating the interaction radius as a function of anode signal
amplitudes, describe detailed simulations to predict the experimental performance, then
present experimental data and demonstrate the ability of radial position sensing to
improve spectroscopic performance.
5.2 Coplanar Anode Position Sensing Theory
To determine the radial position of the interaction within the cylindrical detection
volume, let us examine the sum of the collecting and noncollecting anode weighting
potential distributions within the detector, ( ), ,sum r zϕ θ . To proceed with an analytical
solution, the following assumptions are used:
1. axial variations in ( ), ,sum r zϕ θ are negligible,
2. azimuthal variations in ( ), ,sum r zϕ θ are negligible,
3. the complex anode structure is replaced with a single cylindrical surface defined
by the radius at which the actual anode wires are located.
The axial variations in the weighting potential can be considered negligible within
the active volume of the HPXe detector because most of the change occurs outside the
active HPXe volume, as demonstrated in Chapter 3. Assumption 2 is justified because
the weighting potential boundary conditions are now 1sumϕ = on all anode surfaces and
0sumϕ = at the cathode, so the weighting potential should only have significant azimuthal
variations very near the anode wires. Assumption 3 follows from the prior supposition,
and is necessary to fully define the boundary of the region of interest. The governing
equation is now simplified to
2 2
2 2 2
=00
1 1 0rr r r r z
ϕ ϕ ϕθ
=
∂ ∂ ∂ ∂⎛ ⎞ + + =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ (5.1)
90
Equation (5.1) can be solved to produce (5.2), which defines the summed anode
weighting potential distribution:
( )ln
ln
cat
sumcat
an
Rrr
RR
ϕ
⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞⎜ ⎟⎝ ⎠
(5.2)
In this equation, catR and anR denote the radii at which the cathode surface and the anode
wires lie, respectively.
( )sum rϕ is a function that increases monotonically between the cathode and
anodes, allowing for a correlation between the anode signal and the interaction radius.
However, the weighting potential is not directly measured: the net induced charge on the
anodes is the known quantity, and a relationship in terms of the induced charges must be
developed. The Shockley-Ramo theorem, discussed in Section 2.1, can be used to
correlate the summed anode weighting potential to the sum of the induced charges on the
anodes, ( )sumQ rΔ , where the parameter q is the amount of drifting charge created in the
gamma-ray interaction, and 0r denotes the initial radius of the charge cloud in the
detector:
( ) ( ) ( )0sum sum sumQ r q r rϕ ϕ⎡ ⎤Δ = − −⎣ ⎦ (5.3)
The electron charge created in the detector is approximated by diffq Q−Δ , which is the
difference of the net collecting and noncollecting anode induced charges, and is fairly
uniform throughout the detector due to proper implementation of coplanar anode design.
Thus, the summed anode weighting potential can be determined as a function of the final
measured amplitudes ( )sumQ rΔ and diffQΔ by rearranging (5.3) and assuming the
electrons are fully collected at one of the anodes:
91
( ) ( )0
1
sumsum sum an
diff
Qr RQ
ϕ ϕ=
Δ= −
Δ (5.4)
Equations (5.2) and (5.4) can now be equated to derive the radial coordinate of the
interaction, 0r , in terms of the sum and difference of the measured anode signals, sumQΔ
and diffQΔ :
1
0
sumdiff
cat
cat an
r RR R
⎛ ⎞Δ −⎜ ⎟Δ⎝ ⎠⎛ ⎞= ⎜ ⎟
⎝ ⎠ (5.5)
As with traditional depth sensing in coplanar-anode CdZnTe detectors, this
position-sensing technique relies upon a measured signal that is the product of the
number of charge carriers and a function that is monotonically decreasing with radius:
( )sumQ q f rΔ = − ⋅ . The position dependence is isolated by dividing this signal by the
measured anode signal difference, which is approximately equal to the number of charge
carriers. Near the cathode wall, it is true that sum diffQ QΔ Δ , and equation (5.5) gives the
result 0 catr R= (as it should). For events occurring near the anode wires, 0sumQΔ ,
leading to 0 anr R= , as expected. As we will see later, the only problematic region in the
chamber is in the center of the anode wire structure, for in this volume 0sumQΔ and
equation (5.5) predicts 0 anr R= , introducing an error between the calculated and true
interaction locations. However, the number of events occurring in this region is rather
small, and due to the low electric field most events do not register properly due to severe
charge recombination and ballistic deficit.
Unlike traditional depth sensing, the anode sum provides the position information,
not the cathode. In theory the cathode signal could be used to obtain equivalent radial
information in the HPXe detector. In practice, the use of the cathode signal is limited by
the overwhelming electronic noise, a result of the large cathode-vessel capacitance—
predicted to be 601 pF by the Maxwell 3D simulations.
92
It is important to also quantify the position resolution of the detector. To start, let
us assume the anode sum and difference signal noise terms are uncorrelated, for then the
error propagation formula can be used:
0
222 2 20 0
r sum diffsum diff
r rQ Q
σ σ σ⎛ ⎞⎛ ⎞∂ ∂
= + ⎜ ⎟⎜ ⎟ ⎜ ⎟∂Δ ∂Δ⎝ ⎠ ⎝ ⎠ (5.6)
In (5.6), sumσ , diffσ , and 0r
σ are the standard deviations in the distribution of the
measured sumQΔ , diffQΔ , and 0r , respectively. Using equation (5.5) it is straightforward
to derive the following relationship for 0r
σ :
0
2
2 2
0
ln cat
r an sumsum diff
diff diff
RR Q
r Q Qσ
σ σ
⎛ ⎞⎜ ⎟ ⎛ ⎞Δ⎝ ⎠= + ⎜ ⎟⎜ ⎟Δ Δ⎝ ⎠ (5.7)
Let us assume sumσ and diffσ are approximately equal. If the noise measured on
each anode is independent of the other anode this is a valid assumption, since the error
propagation formula applied to the sum and difference of the two raw anode signals will
then yield the same uncertainty result for both sumσ and diffσ . Let us also convert from
standard deviation to FWHM. The position resolution can now be expressed as a
function of the measured energy resolution,
0
2
0
ln 1r diffcat sum
an diff diff
FWHM FWHMR Qr R Q Q
⎛ ⎞⎛ ⎞ Δ= + ⎜ ⎟⎜ ⎟ ⎜ ⎟Δ Δ⎝ ⎠ ⎝ ⎠
(5.8)
Examining (5.8), the position resolution is expected to vary as a function of
radius. Events occurring near the anodes will induce a summed signal 0sumQΔ , so the
position resolution will simply be the energy resolution multiplied by the geometrical
constant. On the other hand, events near the cathode will be identified by sum diffQ QΔ Δ ,
93
so the position resolution will broaden by a factor of 2 relative to near-anode events.
Assuming the measured energy resolution in the HPXe chambers is about 6% FWHM at
662 keV, and knowing 4cat anR R= , the position resolution is about 8% for events near the
anodes and nearly 12% for events occurring near the cathode. This exercise shows the
importance of improving the measured energy resolution as much as possible.
Additionally, it is evident that the absolute position uncertainty, 0r
FWHM , is not only a
function of deposited energy but also interaction location; as the interaction radius moves
outward toward the cathode, 0r
FWHM becomes larger not just due to the ratio of signals
increasing, but also due to the interaction radius 0r increasing. Thus, it is conservative to
set the radial bin width according to the position uncertainty near the cathode, as the
uncertainty at smaller radii will be noticeably less than this value.
5.3 Simulations
It is important to simulate the detector response as accurately as possible to
predict the performance of the HPXe chambers. These simulations are more
sophisticated than those used during the design phase to optimize the anode structure,
relying upon several different modules to create an accurate representation of the
experiment. The detector simulations incorporate Monte Carlo methods with
electrostatic simulations and knowledge of fundamental HPXe gas properties to create
the best possible model of the detector response. These physical processes are included
in the simulations:
• stochastic particle transport;
• energy deposition via Compton, photoelectric, and pair production processes;
• Fano statistics;
• electron cloud distribution;
• charge recombination along delta-ray tracks;
• electron transport along the established field lines;
• summed and subtracted preamplifier signal generation as a function of time;
• Gaussian filtering of the preamplifier signals;
• superposition of responses for multiple-site events;
94
• sampling from a realistic (experimentally measured) system noise term; and
• calculation of signal amplitudes and the corresponding event energy and radius.
These processes were divided among three different computer codes: Maxwell 3D
to solve the electrostatic fields; a Matlab [95] program developed to simulate the
measured shaped signal as a function of starting position within the detector, given the
HPXe density and biasing conditions; and Geant4 to perform particle transport and
calculate the measured spectra given the energy deposition sequence for each gamma
ray’s history.
It is worth noting that this computational effort seems to be the most detailed
study of HPXe ionization chamber performance, based upon information in the open
literature. Smith, McKigney, and Beyerle attempted similar modeling using Geant4 to
generate ionization densities and locations, they then simulated transport of the electrons
in each event to an unshielded anode in a cylindrical configuration [55]. This particular
model did not include charge recombination, assumed the electric field was a function of
radius only, did not implement a shaping filter, and assumed zero electronic noise. The
results were not particularly accurate, reflecting the simplified modeling.
5.3.1 Maxwell 3D Electrostatic Simulations
The Maxwell 3D electrostatic solver has been used to simulate the operating
electric field and weighting potential distributions within the detection volume. The
modeled geometry is as accurate as possible, incorporating the pressure vessel, all Macor
insulating components, the anode wires, and the cathode surface. The results of these
electrostatic simulations have been written to output files, listing the electric field and
weighting potential at each node on a grid of spacing (0.2 mm, 0.2 mm, 10.0 mm). The
grid spacing in the axial direction can be courser because of the more gradual changes in
the weighting potential and electric field along the axial direction. The spacing in the x-
and y-directions has been chosen to provide sufficient accuracy. By using the actual
Maxwell 3D simulation results, the exercise of finding an accurate Fourier series
representation of the fields is rendered unnecessary; in addition, conditions in the end
volumes where axial changes are non-negligible can be considered in this method.
95
5.3.2 Matlab Waveform Generation Code
The Maxwell 3D results have been used, along with electron-ion recombination,
the electron drift velocity, and electron cloud dimensions to simulate preamplifier
waveforms in a custom Matlab code. These preamplifier signals are shaped using a
Gaussian filter, and the final shaped waveform is saved to a file. The idea behind this
program is that simulated waveforms can be generated for each initial interaction point
on a pre-defined grid; the signals are generated using user-defined time steps. After
completion, the output file contains the simulated waveform at every time step for each
starting location, which can be used by Geant4 to more accurately predict the measured
signals for each gamma-ray event. A grid of starting points with grid spacing (0.5 cm,
0.5 cm, 10.0 cm) is fine enough to properly estimate measured pulse heights without
creating unnecessarily-large data files. A sufficient time step to ensure smooth
trajectories near the anode wires is 100 ns.
Electron-ion recombination information is from Bolotnikov and Ramsey [96],
who measured the fraction of charge extracted from an ionization cloud for different
combinations of gamma-ray energies, HPXe densities, and electric field magnitudes.
Equation (5.9) fits the published data for 137Cs photoelectric absorptions in HPXe gas of
density 0.3 g/cm3; the electric field magnitude E has units [kV/cm]:
( )0
14.65 0.228 0.016log
QQ E
=−
(5.9)
The electron cloud is assumed to be a sphere of diameter 3.65 mm, which is the
mean diameter calculated using Geant4 simulations for a 662-keV photoelectron in 0.3
g/cm3 HPXe gas [97]. In these simulations, monoenergetic photoelectrons are directed
randomly into the Xe gas: as they create ionizations, the position of each interaction is
compared to previous coordinate maxima and minima, and these extrema are updated if
necessary. The electrons are tracked until their mean free path reaches a user-defined
value of 1 μm. The impact of this cut value was studied by decreasing it to 0.1 then 0.01
μm, but the resulting cloud distribution did not change. The results of this cloud diameter
calculation are presented in Figure 5.1.
96
The true electron cloud dimension will generally be smaller when less energy is
deposited in an interaction, but for the simulations only a single cloud diameter is used.
This assumption should slightly worsen the results in comparison to the experiment, as
electron arrival at the collecting anode will appear more dispersed in time, leading to
greater pulse height deficit. Generally a minimum of 500 electrons are dispersed
randomly throughout the cloud to transport through the detector for signal generation
purposes. This number was found to reduce the error arising from fluctuations in the
final induced charge to well below 1%.
Figure 5.1. Geant4 simulated distribution of the charge cloud diameter in the HPXe detector for 662-keV depositions.
Based on the electron drift speed data published for HPXe at density 0.6 g/cm3 by
Ulin et al. [48], the following simple relationship is used in the simulations to determine
the drift speed at each time step:
97
; 1.0
1.0 ; 1.0e
kV mm kVE Ecm s cm
vmm kVE
s cm
μ
μ
⎧ ⎡ ⎤ <⎪ ⎢ ⎥⎪ ⎣ ⎦= ⎨⎪ ≥⎪⎩
(5.10)
When consulting published drift speed data, it is important to realize that the drift speed
is a function of E N , where N is the atom density of the gas [98]. Thus, since the
coplanar-anode HPXe chamber has gas density 0.3 g/cm3, a particular drift speed is
realized at half the published field strength in the current chamber.
When an event is generated within the detector, preamplifier sum and difference
waveforms are calculated at each time step over the duration of the pulse length, with the
position of each electron being updated at each point in time given the local electron drift
velocity. The final preamplifier signals are calculated by multiplying the signal at each
point in time by the fraction of electrons escaping recombination, then dividing by the
total number of electrons in the cloud. The final preamplifier pulse is thus normalized to
a value of 1.
To simulate the presence of a Gaussian shaping amplifier, a ( )4CR RC− filter is
implemented; this filter has been chosen because it closely approximates a true Gaussian
distribution [8]. The impulse response function ( )h t used to achieve this filter is given
by the following equation, where n is the number of RC integration stages in the filter
and s RCτ = is the shaping time:
( ) 1 e 1s
n tns
ns s
nth tn t
τ ττ τ
−⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
(5.11)
The convolution of this filter with a preamplifier waveform ( )w t produces a
Gaussian-shaped pulse. First the Fourier transform is used to calculate the frequency
response of both the signal and the filter, which is easy because the Fourier transform and
its inverse are built-in functions in Matlab. Denoting the Fourier transform of the
shaping filter and preamplifier waveform as
98
( ) ( ) ( )
( ) ( ) ( )
e
e
i t
i t
H h t h t dt
W w t w t dt
ω
ω
ω
ω
∞−
−∞
∞−
−∞
≡ =
≡ =
∫
∫
F
F
(5.12)
the definition of convolution and the properties of the Fourier transform [82] are used to
calculate the shaped filter output, ( )S t .
( ) ( ) ( )
( ) ( ) 1
S t w h t d
W H
τ τ τ
ω ω
∞
−∞
−
≡ −
=
∫F
(5.13)
The shaped waveform is written to a data file as a function of time; all of the
waveforms generated for each of the preamplifier sum and difference signals are
contained in these data files, along with the corresponding interaction location.
5.3.3 Geant4 Monte Carlo Simulations
The waveform simulations are combined with Geant4 Monte Carlo simulations to
provide the most detailed model of detector response achievable. In these simulations,
the physical geometry is represented as accurately as possible, and gamma rays are
emitted from a 137Cs point source located on a plane bisecting the detector, 25.4 cm from
the vessel wall. In the code, energy depositions within the sensitive volume of the
detector trigger a sequence of calculations that first records the location and energy
deposited for the interaction. To introduce Fano statistics, the number of charge carriers
produced in the interaction, N , is sampled from a normal distribution characterized by
mean and standard deviation
dep
N
EN
wFNσ
=
=
(5.14)
99
where depE is the energy deposited in the interaction by the gamma ray, w is the mean
energy required to create an ionization, and F is the Fano factor. For these simulations,
21.9eVw = and 0.17F = [35].
The interaction location is mapped to the closest node for which a response was
generated in the Matlab waveform program. This can involve reflections and translations
through the detector geometry, possible due to the symmetry of the design. After each
gamma ray history is complete, the shaped added and subtracted waveforms for each
interaction location are scaled by the number of electrons generated in the corresponding
interaction, and then summation over all interaction locations is executed to create a
single shaped response for both the added and subtracted signals. Superposition is
possible due to the linearity of the filter and the Fourier transforms. From these
waveforms, the maximum amplitude of the added and subtracted waveforms can be
determined; these values are necessary to form the energy spectrum and to calculate the
event radius.
Finally, electronic noise is added to the system by separately sampling from a
normal distribution to determine the collecting and noncollecting anode noise terms.
(This is only valid if there is no correlation in the experimental noise terms.) The normal
distributions are characterized by a mean of 0 and a standard deviation that is shaping-
time dependent, estimated from the measured equivalent noise charge data of Section
4.2.2. The noise-perturbed signal amplitudes are used to generate the measured deposited
energy and the calculated radius via equation (5.5). Energy and radially-separated energy
spectra are generated by placing each event into 1-keV wide energy bins and dividing the
continuum of radial values into 10 bins, which corresponds to the position uncertainty
calculated using equation (5.8).
5.3.4 Simulation Results
The purpose of the first set of simulations is to determine an appropriate shaping
time for a given set of detector biases: -4000 V on the cathode, +1400 V on the collecting
anode, and the noncollecting anode at ground potential. This detector biasing scheme is
typical for experiments, and therefore very relevant. To begin, make the following
assumptions to simplify this set of computations:
100
• the electron cloud has zero diameter;
• there is no axial variation in the operating or weighting fields;
• all interactions are via photoelectric interactions;
• all events create exactly 0N electrons (i.e., 0F = ); and
• there is no electronic noise.
Given these assumptions, shaping times of 6, 10, 16, 20, and 32 μs have been
modeled. The first three shaping times are all experimentally realizable on the current
laboratory equipment; the final time constants are useful to determine the spectrum shape
without ballistic deficit concerns. Events were generated throughout the interaction plane
on a uniformly-spaced grid, and the simulated signals recorded to form a differential
pulse-height spectrum.
Figure 5.2 shows the photopeak region of the simulated spectra, assuming the
incoming gamma rays have energy 662 keV; in this plot, the data series are separated for
clarity by vertical shifts. It is evident from this figure that the cases with 6 and 10 μs
shaping time constants suffer severe degradation in the photopeak due to ballistic deficit.
The 20 and 32 μs time constants seem to be sufficient for avoiding ballistic deficit
broadening. The minor peak near channel 580 in each data series is an artifact of the
weighting potential discretization; several events near the center of the detector are
assigned similar initial weighting potentials.
Figure 5.3 is a scatter plot which displays the interaction radius calculated with
equation (5.5) against the true interaction radius for each event. As the shaping time
constant is increased from 6 to 32 μs, there are two beneficial effects: first, the calculated
radius becomes a better representation of the true radius for near-cathode events; second,
the distribution in calculated radii for a particular true interaction radius becomes
narrower. The combination of these two effects allows a more accurate measurement of
the interaction radius. The reason for this effect is, once again, ballistic deficit at short
shaping times, since the rise time of the summed signal for a near-cathode event can be
near 40 μs. When short time constants are used, the summed signal’s shaped amplitude
is decreased and the exact path of the electron cloud becomes more important, giving rise
to the wider distribution of calculated radii for a fixed interaction radius. Even for the
101
longest shaping time, the measured radius does not reach 1.0 for a near-cathode event—
this is not a problem, since as long as the calculated radius increases monotonically with
the true radius, a correlation can be made.
In Figure 5.3 there is a plateau at a calculated radius of about 0.04. Due to the
nature of the summed anode signal’s weighting potential, events occurring anywhere
within the anode structure will always have nearly uniform summed anode responses. In
addition, near the anodes, which are centered at radius 12.7 mm, there is a fairly large
variation in calculated radii due to the dramatic change in the anode weighting potential
difference, which varies from -1 to +1 between the noncollecting and collecting anode
wires, respectively.
Figure 5.2. Simulated effect of shaping time choice on the photopeak region. Vertical offsets are used to separate the data.
102
From the figures, it seems that a minimum shaping time of 20 μs should be used
to improve both energy and position resolution. In practice, the energy resolution will
also be affected to some degree by the inclusion of all of the assumptions listed at the
beginning of the section; electronic noise will certainly play a major role, favoring shorter
shaping times than 20 μs. (For example, in Chapter 4 the best energy resolution was
measured using a 12 μs time constant.) The best course of action may be to choose two
shaping time constants: the preamplifier difference signal will be shaped with a filter that
minimizes photopeak width, while the preamplifier sum signal is shaped with a longer
shaping time that minimizes ballistic deficit.
Figure 5.3. The calculated interaction radius plotted against the true interaction radius for several shaping time constants. (Longer shaping times correspond to greater calculated radii.)
103
Let us continue with shaping time constants of 20 μs for the anode sum and
difference signals. It is time to consider a simulation that uses realistic approximations at
all physical parameters of significance. The assumptions made at the beginning of this
section have been replaced with the following:
• the electron cloud has diameter 3.65 mm;
• axial variation in the operating and weighting fields is accounted for by putting
each interaction into one of five axial slices to calculate the detector response (3
slices in the central region centered at z = 0, 25, and 50 mm; 2 slices in the end
region centered at z = 68 and 78 mm);
• interactions can be photoelectric absorptions, Compton scatters, or pair
production as dictated by Geant4’s cross-section library;
• a Fano factor 0.17F = is used to simulate statistical broadening; and
• electronic noise is extrapolated from experimental data to an approximate value of
450electronsENC = for the total system noise.
The simulation results are presented below. To obtain good statistics, the Geant4
run considered 20x106 662-keV photons emitted from a point source located 25.4 cm
from the outer wall of the vessel and centered axially. The energy spectrum is plotted in
Figure 5.5, and is labeled the raw spectrum. The measured energy resolution of this
spectrum is 4.6% FWHM.
It is possible to use the radial sensing technique to plot the simulated energy
spectrum as a function of radius, as done in Figure 5.4 with 10 radial bins spanning the
physically-meaningful range of calculated radii, one for negative calculated radii, and one
for calculated radii that are too large to be correct. There are several important features
in this spectrum:
1. There is basically no response at calculated radial bins 0, 1, and 2. These radii
physically lie inside the anode structure, and events in this volume are shifted to
larger radii (see Figure 5.3).
104
2. As the radius increases, the photopeak exhibits improved energy resolution, with
dramatic changes occurring in radii 3-5 (near the anode wires). This is attributed
to weighting potential nonuniformity in the immediate area.
3. As the radius increases, the integral number of counts measured also increases, as
expected in a cylindrical geometry. The maximum number of counts appears in
radial bin 8; consulting Figure 5.3, this can be explained because bin 8 is the
largest bin for which the full range of calculated radii can be achieved.
4. Radial bin 11 contains mainly low-energy events, but no classic gamma-ray
spectral features. This is because the radius lies physically outside the chamber; it
corresponds to events with the anode signal sum larger than the difference, which
happens mainly when electrons are improperly collected at the noncollecting
anode, thereby reducing the anode difference signal but not the sum.
5. Although it is difficult to see in this plot, the photopeak centroid channel varies
slightly as a function of radial bin. This is due to variations in ionization cloud
charge recombination at large radii; at small radii, the nonzero weighting potential
is a significant contributor.
Figure 5.4. The radially-separated energy spectrum for the 137Cs simulation. Radial bin 0 theoretically corresponds to the center of the chamber, and radius 10 is at the cathode.
105
Some of these observations can be used to improve the quality of the
measurements. For example, events recorded in radii that are physically meaningless can
be neglected because they are a result of improper charge collection. In addition, events
in radii 3 and 4 can be neglected because the energy resolution is very poor in these
regions of the detector. For the remaining radii, a radial bin-dependent gain can be
applied to align all photopeak centroids in the same channel (arbitrarily chosen as
channel 662). The results of these corrections to the raw data are demonstrated in Figure
5.5. In the figure, it is evident that the ratio of photopeak to Compton continuum counts
is noticeably improved, and the measured energy resolution improves from 4.6% to 4.4%
FWHM at 662 keV. These enhancements are useful for standard spectroscopic
measurements.
Figure 5.5. The effect of radial corrections on the simulated 137Cs energy spectrum. The energy resolution of the photopeak is improved from 4.6% to 4.4% FWHM.
106
5.4 Experiments
To perform radial position sensing, both the sum and difference of the collecting
and noncollecting anodes are required. These signals are generated using simple addition
and subtraction circuits included in the preamplifier shielding box; the output of each
circuit is filtered using a shaping amplifier, and these shaped pulses are sent to a peak-
hold circuit, where the maximum amplitudes of both signals are held for a fixed period of
time after the circuit is triggered. The peak-hold circuit generates a trigger that signals
the data collection system to sample the peaks before they are discharged. The data
collection is performed via a LabVIEW [99] program recording data from a PCI-6110
data acquisition card. A schematic of this system is shown in Figure 5.6.
Figure 5.6. A connection diagram of the detection system used for radial position sensing measurements.
The data acquisition system digitizes the incoming signals with 12-bit accuracy,
thereby partitioning the incoming signals spanning a range of ±10 V into 4096 channels
[100]. The shaped anode sum and difference amplitudes are written to a data file, each
row representing a recorded event. The data post-processing is accomplished using
Matlab.
Using the HPXe optimization study presented in Chapter 4 as a reference, data
was collected using the following system parameters: the cathode and collecting anode
were biased at -4500 V and +1400 V, respectively; a 137Cs source was collimated to
direct source photons to the central plane of the detector; the subtracted signal was
filtered using triangular shaping with a 12 μs shaping time constant; and a counting time
HPXe Detector
CA Preamp
NCA Preamp
Cathode HVPS
CA HVPS
Subtr. Circuit
Shaping Ampl.
DAQ
Sum Circuit
Shaping Ampl.
Peak Hold
107
of 1800 s was used, with background counted alone for an equivalent time. The summed
signal was shaped with a Gaussian filter using the longest available shaping time—16
μs—in order to minimize ballistic deficit.
The data is presented as a function of radius in Figure 5.7 and Figure 5.8. In these
plots, the radius refers not to the theoretical radius calculated using equation (5.5), but
instead to the ratio of the measured signal amplitudes of the anode sum and difference
signals, ξ :
2
1
sum
diff
diffsum
diff diff
FWHMQFWHMQ Qξ
ξ Δ=
Δ
⎛ ⎞Δ= + ⎜ ⎟⎜ ⎟Δ Δ⎝ ⎠
(5.15)
This expression is sometimes more convenient to work with because the FWHM in the
distribution of ξ measured for a particular energy is slowly varying with respect to
radius, whereas in equation (5.8) the position resolution is slowly varying, meaning the
absolute width of the bins can change quite a lot between the anode and the cathode.
Assuming the energy resolution is around 6%, [ ]0.060,0.085FWHMξ ∈ . The induced
charge ratio is physically expected to span the range [ ]0,1ξ ∈ , so 10 bins spanning this
range is an appropriate number given the limitations on resolution. Figure 5.7 shows the
data including source counts along with background and a test pulse; in Figure 5.8 the
background has been stripped out. When comparing these two plots it is evident that
background events occur mainly at low energies, and are fairly evenly distributed
throughout the entire chamber.
108
Figure 5.7. A radially-separated experimental 137Cs energy spectrum. The radial index is simply the anode sum-difference ratio in this plot. Increasing indices correspond to increasing physical radii. A test pulse is present in radial index 9 near channel 850.
Figure 5.8. An experimental 137Cs radially-separated spectrum with background stripped out, but otherwise identical to Figure 5.7.
109
Examining Figure 5.8, effects are observed that are similar to the simulation
results presented in Figure 5.4, with a few notable exceptions: there are not empty small
radial bins, which is due to the difference in the radius calculation method; many of the
large radial bins are nearly empty, a ballistic deficit effect caused by relatively short
shaping time constants compared to the summed signal rise times; and the final
difference is the inclusion of a test signal which shows as a peak near channel 850 in
radial bin 9. The experimental data does exhibit very broad or nonexistent photopeaks at
small radii, just as the simulations predicted: this effect is due to poor weighting potential
uniformity near the anodes. As the radius increases, so does the integral number of
counts, which is consistent with the cylindrical geometry. Radial bin 11 contains only
low-energy counts, also predicted by the simulations, resulting from improper charge
collection. In addition, it is true that the photopeak centroid shifts slightly as a function
of radial index, due to changes in the weighting potential distribution and charge
recombination.
Let us consider using the radial information to improve the energy spectrum, as
was done with the simulations. First, let us discard data from radii that do not contain
discernable photopeaks, as these channels only degrade the spectrum. This leaves only
radii 3 through 6 as desirable data. Now, apply a gain to each radial bin that aligns each
centroid at channel 662, an operation which reverses the drift in photopeak centroid as a
function of radius. After re-processing the data to filter out the unwanted events and
apply an appropriate gain to the remaining data, the overall energy spectrum is noticeably
improved. The new spectrum is presented in Figure 5.9 along with the original data for
comparison. The measured energy resolution improved from 5.9% to 5.5% FWHM at
662 keV, which is a statistically-significant improvement considering the uncertainty in
each resolution measurement is calculated to be about 0.03% [8]. In addition, it is
obvious that the photopeak-to-total count ratio is greatly increased. The measured noise
limit is 4.2% FWHM, which when subtracted in quadrature gives an intrinsic detector
resolution of 3.6% FWHM. This value is much greater than the resolution limit predicted
by Fano statistics, and will be analyzed in Chapter 6 for sources of degradation and
possible improvements.
110
Figure 5.9. The effect of radial corrections on the experimental 137Cs data. The photopeak resolution improved from 5.9% to 5.5% FWHM as a result of the corrections.
111
CHAPTER 6
CONSIDERATIONS FOR IMPROVING PERFORMANCE
6.1 Energy Resolution Enhancement
Previous chapters have demonstrated performance of the coplanar-anode HPXe
detectors as good as 5.5% FWHM for a collimated 137Cs source. This represents a
significant advancement compared to previous coplanar-anode HPXe designs, which
achieved just 9% FWHM at 662 keV [60]. Nevertheless, to become a viable alternative
to gridded HPXe chambers the energy resolution must improve to between 3.5 and 4%
FWHM, which represents common performance for a large-diameter gridded chamber.
The purpose of this chapter is to investigate physical processes important in signal
formation, quantifying the contribution of each process to the overall measured peak
width. This study will suggest how improvements can be made, and at the same time
limits on performance improvement may become apparent.
6.1.1 Simulation Methods for Physical Process Contributions
One of the nice features of the Maxwell 3D/Matlab/Geant4 simulation package
described in Chapter 5 is that it lends itself to quantifying the effect of each physical
process upon the measured photopeak width. Since physical processes are introduced
one-by-one, the desired results are obtained by creating tallies of the measured pulse
amplitudes after each physical process is added to the model. By creating several energy
spectra in this manner, it is possible to quantify the effect of each physical process by
comparing the photopeak FWHM before and after each step of physics implementation.
Starting from the true energy deposition spectrum, the physical processes that are
quantified are
112
• Fano charge carrier statistics,
• weighting potential distribution at the interaction location,
• charge recombination at the ionization site,
• nonzero electron cloud radius,
• improper electron collection due to insufficient anode biasing,
• pulse shaping,
• electronic noise, and
• axial field nonuniformities.
The effects of pulse shaping are split into two parts: a geometrical portion and a
contribution due to timing. The geometrical factor considers the slightly different shaped
amplitudes resulting from interactions in different sections of the detection volume.
These differences arise mainly from the variations in the subtraction circuit pulse shape.
The timing factor acknowledges that due to the potentially long drift time of electrons in
this system, a multiple-interaction event can suffer from pulse-height deficit. This deficit
arises because the response of the shaping filter peaks at different times for the different
charge clouds created in the detector, and the time offset means the measured pulse
amplitude is slightly less than the sum of the responses when each cloud is considered
individually. Figure 6.1 demonstrates this effect using simulated data and a ( )4CR RC−
shaping filter with a 12-μs time constant. The response to single-site events creating a
single electron at (18 mm, 0, 0) and (48 mm, 9.5 mm, 0) are shown in black and blue,
respectively. The response for a two-site event creating a single electron at each of these
two locations is shown in red. In this case, 2 units of charge are created; the amplitudes
of the filter response to each drifting electron are 0.9948 and 0.8812, the deviations from
unity caused mostly by weighting potential. Thus, the ideal measured amplitude would
be 1.8760 units of charge, but in fact the time delay causes the true two-site event to have
a measured amplitude of 1.5046 units of charge, a deviation of 19.8%.
113
Figure 6.1. The response of the shaping filter to events at (18 mm, 0, 0) and (48 mm, 15 mm, 0) are shown in black and blue, respectively. The response to a two-site event is shown in red and suffers from pulse height deficit.
Some inaccuracies in the results are expected due to limitations of the current
simulation package. For example, the electron cloud is not treated stochastically, but
instead a single fixed cloud distribution is used at each interaction location. This
approximation will reduce the observed variance in the simulation results. In addition,
discretization of the continuous weighting potential and electric field distributions may
introduce some inaccuracies, although if the discretization is fine enough this error term
may be negligible. Finally, processes such as ion drift and charge diffusion are not
modeled currently, while the model used for charge recombination statistics is only
assumed to be correct. Positive ions are expected to have a negligible effect on the
measured signals—after all, coplanar anodes are supposed to remove sensitivity to ion
motion—but electron diffusion is an important term that can affect charge collection
114
times and the fraction of the generated electrons terminating their paths at the
noncollecting anode. Recombination statistics are considered to be Poisson in nature,
which assumes that individual recombination events are independent of one another and
each have small probability of occurring. Although the simulation package will not be
revised, it is important to recognize the limitations of the model.
6.1.2 Spectral Contributions in Pure Xe with a 20 μs Shaping Time
Let us first consider the case with ideal shaping times in pure Xe gas, as
determined by the simulation results in Figures 5.2 and 5.3. In this instance, the ideal
shaping times are defined to be those which minimize ballistic deficit problems, and the
simulations showed 20 μs shaping time constants to be sufficient for both the anode sum
and difference signals. The parameters of interest in this simulation are listed in Table
6.1.
Table 6.1. A list of important parameters in the energy resolution study.
Parameter Value
Gas density 0.3 g/cm3
Cathode/collecting anode bias -4000 V / +1400 V
Electrons per cloud 600
Electron cloud diameter 3.65 mm
Simulation mesh spacing (0.5 mm, 0.5 mm, 25.0 mm)
Shaping time constants 20 μs
Shaping filter ( )4CR RC−
Waveform time step 100 ns
Simulated source 137Cs point source
Number of Geant4 primary particles 20x106
Mean ionization energy 21.9 eV
Fano factor 0.17
ENC (extrapolated from measured data) 450 electrons
115
The results of these simulations follow. Let us begin with three spectra that
cannot be changed with bias or shaping time: the true energy deposition spectrum, and
then this spectrum broadened first by Fano carrier statistics and then by the weighting
potential distribution at the gamma-ray interaction locations. Furthermore, the results
will be idealized at this stage by considering only two-dimensional fields— ( ),wp rϕ θ and
( ),r θE —with no variation in the axial direction. These spectra are shown in Figure 6.2,
with the left panel showing the entire spectrum and the right zooming in on the
photopeak region. It is evident that the two physical processes broadening the true
deposition account for very little photopeak broadening.
Figure 6.2. A comparison of the true deposited energy spectrum to the spectrum broadened by Fano carrier statistics, then by the weighting potential distribution at the interaction location.
Let us now consider the effects of charge recombination, electron cloud
dimension, and the effects of nonideal anode biasing upon electron collection. For ease
of comparison to previous results, the final tally from Figure 6.2 is carried over to Figure
6.3. It is evident that charge recombination is an important physical process, not only
because of photopeak broadening but also because it reduces the measured pulse
amplitude. In fact, while some of the broadening effect is statistical in nature, a part is
due to the magnitude of charge recombination as a function of radial coordinate, since
this effect depends upon the local field strength. This particular broadening term should
116
be mostly compensated by photopeak alignment when radial position sensing is used, and
can be reduced without photopeak alignment by increasing the strength of the electric
field throughout the chamber.
Figure 6.3. A comparison of the energy spectrum broadened by Fano statistics and the weighting potential distribution to the spectra when charge recombination, electron cloud distribution, and improper collection due to nonideal anode biasing are considered.
The distribution of electrons in a cloud at the ionization site shows negligible
effect upon the spectrum in Figure 6.3; as mentioned before, this simulation may
underestimate the effect of the cloud size due to the fixed cloud structure simulated. The
effect in this particular simulation is to average the weighting potential over the cloud
volume, so in regions where the weighting potential distribution exhibits little curvature
the simulation of an electron cloud is expected to have no effect.
Finally, improper collection of electrons at the noncollecting anode is seen to
reduce the number of photopeak counts, although there is not an apparent effect on the
photopeak width. The count reduction effect is observed experimentally, and changes
with collecting anode bias: see Figure 4.10 for one example. The collecting anode bias
was known in advance to be insufficient after studying Maxwell 3D simulations of the
operating electric field distribution for a cathode bias of -4000 V. Figure 6.4 shows the
electric field lines near the anodes along the central axial plane. In panel (a) there are
field lines that originate at the noncollecting anode wires, which are surrounded by blue
and green coloring, when the collecting anode bias is VCA = +1400 V. This implies that
117
electrons can be collected at these wires, since electron drift opposes the direction of field
lines.
(a)
(b)
Figure 6.4. Examining the electric field lines along the central plane near the anodes when Vcat = -4000 V; collecting anode wires are surrounded by red. (a) VCA = +1400 V is not sufficient to keep electrons from terminating on the noncollecting anode wires. (b) VCA = +2000 V ensures ideal collection of electrons.
118
In panel (b) of Figure 6.4, the collecting anode bias is raised to +2000 V. Now
there are no field lines that originate at the noncollecting anode, and an electric potential
saddle point can be seen in the dark blue region just right of the noncollecting wire. This
indicates that no electrons will pass through that region (as discussed in Section 2.2.3),
but will instead be redirected toward the collecting anodes at the top and bottom of the
figure.
Returning to Figure 6.3, improper anode biasing actually acts to slightly reduce
the photopeak width. As will be seen later, events occurring directly outward from a
noncollecting anode tend to induce a slightly larger signal than those starting closer to a
collecting anode wire, so reducing the electron collection does not affect the low-energy
side of the pulse-height distribution, but it does drop the events at the high-energy side to
lower channels, and thus as long as only a small fraction of electrons are improperly
collected, the effect will be to decrease the photopeak width. Events that have a great
deal of overlap with a noncollecting anode wire are completely removed from the
photopeak, which accounts for the reduced number of observed counts in the figure.
Referencing Equation (2.8), the quantitative effect of improper charge collection is that
every electron terminating at the noncollecting anode carries a weight of -1 instead of +1,
so each improperly-collected electron actually reduces the measured amplitude by a
weight of 2. Thus, events with substantial electron collection at a supposed noncollecting
wire end up far outside the photopeak, and may appear in a negative MCA channel.
The next effect to be studied is that of the Gaussian filtering. Figure 6.5 displays
two spectra of shaped anode difference signals: one adds the shaped maximum of each
individual response for every interaction in multi-site events to obtain the shaped
amplitude, and this spectrum is labeled “geometry” because it represents the change in
the response of the shaping filter to events throughout the geometry. The other spectrum
accounts for time-of-arrival differences in multi-site events, which reduce the overall
amplitude because the individual maxima do not overlap (see Figure 6.1); this spectrum
is labeled “timing.” For comparison to Figure 6.3, these two spectra are plotted along
with the spectrum of anode difference amplitudes when improper electron collection is
considered. In Figure 6.5 it is evident that the geometrical response of the shaping filter
119
only broadens the photopeak by a small amount, while timing adds in a small, but
noticeable, low-energy tail on the photopeak in addition to further broadening.
Figure 6.5. Examining the effects of shaping upon the energy spectrum. The shaping process is divided into two portions: a geometrical part and a time-of-arrival part.
In the final set of plots, the spectrum of events after shaping is first broadened
with electronic noise; the effects of axial field nonuniformity are added next by using
better approximations to the local field values, not simply projecting the distribution at
the central plane throughout the entire chamber. Finally, radial position sensing is
implemented to align photopeaks as a function of calculated radial location. It is obvious
from this figure that electronic noise is a major contributor to the overall peak
broadening. Axial field nonuniformity is also an important contributor, not only because
it broadens the photopeak even further, but because it also results in a large loss in
photopeak counts. This is due to the end gas spaces on either end of the chamber, where
the relatively weak electric fields result in stronger recombination and also large ballistic
deficit due to the long signal rise time compared to the shaping time constant.
120
Figure 6.6. Comparing the shaped spectrum without noise, labeled “Shaping,” to that when noise, axial field nonuniformity, and photopeak alignment are considered. The figure on the right enhances the photopeak region to better display differences.
There is another subtle effect produced by the axial field nonuniformity, which is
an upward shift in the photopeak centroid by several channels. To investigate this, the
previous set of simulations was repeated, but instead of projecting the field distributions
from the central plane throughout the entire HPXe detector volume, this time the
distribution used came from a plane offset from the center by 48 mm, which is near the
end of the central detection volume. The results are shown in Figure 6.7. In the figure it
can be seen that the weighting potential distributions from the two planes are nearly
identical. The effects of charge recombination are radically different, though, with the
plane offset by 48 mm exhibiting a much broader distribution of charge loss due to
recombination, although the centroid of this distribution is observed to be between 15 and
20 channels higher than the more uniform distribution from the central plane.
Apparently, the electric field near the edge of the central volume is measurably stronger
in magnitude and also less uniform throughout the plane. This difference in
recombination explains the axial effects observed in Figure 6.6, and will be observed
experimentally in Chapter 7.
Returning to Figure 6.6, it is possible to see the beneficial effects of the photopeak
alignment process upon the measured peak FWHM in the right panel. In the left panel,
the reduced Compton continuum can be clearly observed.
121
Figure 6.7. A comparison of the simulation results when the field distribution projected throughout the entire volume corresponds to the planes z = 0 or z = 48 mm. The weighting potential distributions from these two planes are nearly identical.
The physical processes impacting the measured energy resolution in a spectrum
have so far been discussed only in a qualitative manner. To quantify the effects that have
been observed in the preceding figures, let us assume that each process, when acting upon
an input distributed evenly in space and as a delta function in energy, produces a
normally-distributed spectrum with an associated FWHM. Since these processes are
Gaussian, the FWHM of each one sums in quadrature to produce the FWHM measured in
the final spectrum. By simply measuring the FWHM of each spectrum tallied, the
FWHM of each process can be calculated easily with the aforementioned assumptions.
FWHM measurements were performed by importing the data into EG&G ORTEC’s
MAESTRO software, which has built-in peak fitting algorithms; the final results are
shown as a histogram in Figure 6.8 in ascending order of the process’ FWHM. The total
measured FWHM is also shown in red for reference.
122
Figure 6.8. A comparison of the FWHM attributed to each physical process studied in the simulations. Van refers to improper anode bias and the resulting incomplete charge collection; the effects of the shaping filter have not been split into separate components.
In the figure it is obvious that electronic noise is by far the largest contributor to
the measured photopeak FWHM. Other contributors to the measured FWHM that would
ideally need to be reduced to produce an excellent spectrometer are the axial field
variations and the charge recombination, which is actually slightly more important than
the histogram conveys because the associated large downshift in centroid channel is not
taken into account here. Terms that reduce the measured FWHM—charge recombination
and photopeak alignment—are represented in the histogram by negative FWHM values,
although technically these terms are imaginary when using the method of summing
component terms in quadrature.
123
Table 6.2. A summary of important results from the physical effects study.
Process FWHM (channels) Centroid (channels) Normalized peak area
True deposition 0 662 1.000
Fano statistics 3.74 662 0.993
Weighting potential 2.28 662 0.785
Recombination 6.77 621.7 0.736
Cloud distribution 0.40 621.7 0.739
Improper collection (2.93i) 621.8 0.555
Shaping, geometry 3.34 621.3 0.550
Shaping, timing 2.50 621.0 0.534
Electronic noise 24.74 621.0 0.586
Axial nonuniformity 10.37 624.6 0.468
Peak alignment (8.03i) 625.4 0.456
Table 6.2 lists the important quantitative parameters for each physical process: the
process’ FWHM, the measured photopeak centroid after the process is tallied, and the
normalized peak area after the process is included. The only significant shift in centroid
location comes from charge recombination, although the effect of axial field
nonuniformity can be seen. The normalized peak area includes some uncertainty that
depends on the background stripping algorithm and the placement of the region-of-
interest boundaries; still, it is evident that the improper anode biasing and axial
nonuniformity have real effects on the measured number of photopeak counts. The
weighting potential distribution also seems to reduce the photopeak area significantly by
distributing counts from the anode region far beyond the photopeak.
6.1.3 Analysis of Events Along an Arc of Fixed Radius
One question that arises from the preceding study is: how does electron
termination at the noncollecting anode decrease the distribution of signal amplitudes
produced by the anode differencing circuit? A hypothesis was proposed in the previous
section without any analysis; the question will now be studied.
124
To investigate this question, let us assume the same operational conditions used in
the previous simulations—i.e., the same source, electrode biasing, and shaping filter
conditions. Let us fix the interaction radius at a large value, 40 mm, and simulate
interactions at 0.1° intervals between 0 and 30.0°; the electric field and weighting
potential should be very uniform this far from the anodes, so weighting potential and
recombination differences will be negligible. The simulations described in the previous
section are now repeated, and the results displayed in Figure 6.9 as a function of
azimuthal angle. For reference, the collecting anode is centered at azimuth 0°, the
neighboring noncollecting wire at 30°. The right panel in the figure expands the range
near ordinate unity for a better view of functional behavior.
Figure 6.9. Investigating the effect of azimuthal angle on the distribution of signal amplitudes. The collecting anode is centered at 0°, the noncollecting anode at 30°. The right panel zooms in on the region around unity ordinate for clarity.
On the left panel, it is obvious that recombination is nearly constant for all of the
points considered; the weighting potential difference at the interaction point (black) and
averaged over the entire cloud (green) also appear to be constant with angle. The effect
of improper electron collection can be seen in the blue data series: the distribution is
fairly uniform to the eye between azimuths 0 and 26°, but then falls of quickly to
negative values, indicating more electrons being collected at a designated noncollecting
anode wire than at the collecting anode. The shaped signal amplitude (red) follows the
125
collected charge closely, although it does not go negative because the method for
measuring the shaped signal amplitude in these simulations is to return the most positive
value, not the largest deviation from the baseline.
Figure 6.10. A plot of preamplifier and shaped waveforms for different azimuthal angles (left). Zooming in on the peak region shows the nonlinear response of the shaping filter (right).
The right panel of Figure 6.9 zooms in to show more subtle behavior. It is now
apparent that the weighting potential difference distribution is not perfectly uniform, but
instead appears to give a slightly larger net induced charge nearer the noncollecting
anode (this assumes perfect collection of electrons, of course, which is not the case).
Still, the weighting potential variation from one extreme to the other is only about 0.2%.
The shaping filter response as a function of angle is more interesting. The filtering
results in an amplitude deficit compared to the net weighting potential difference for
events nearer the collecting anode; as the azimuthal angle increases, the filter response
increases nonlinearly and eventually surpasses the net weighting potential difference.
This indicates that the shape of the preamplifier difference signal impacts the filter
response, as the preamplifier difference signal has a slightly more gradual slope for
electrons directly approaching the collecting anode than those closer to the noncollecting
wires. The difference in preamplifier signal shape is demonstrated for three locations in
126
Figure 6.10: azimuths 0, 12°, and 24°. The imperfect shaping filter response explains the
process’ contribution to the measured photopeak FWHM, as shown in Figure 6.8.
6.1.4 Photopeak Broadening Contributions for Realistic Shaping Times
The studies performed in the preceding sections provide useful information, but
do not use shaping times comparable to the experimental conditions. The simulation of
Section 6.1.2 is repeated here using a shaping time constant of 12 μs, which is used in
most of the experiments performed on the system. This change will not affect the
contributions from Fano charge statistics, weighting potential, recombination, or
incomplete charge collection. It will, however, change the contributions from the
shaping filter, as ballistic deficit will increase. In addition, the system electronic noise
will decrease, as observed in experiments. The ENC used in this simulation is linearly
interpolated between experimentally-measured values: 390 electrons.
The results of this study are displayed as a histogram in Figure 6.11; the data for a
20 μs shaping time is included for comparison. The spectra are not shown because they
are qualitatively very similar to the 20 μs case, just with different peak widths. The
measured photopeak FWHM for the 12 μs case has decreased due to the diminished
electronic noise term, despite increases in the FWHM contributions from shaping and
axial nonuniformities, and less improvement from photopeak alignment. The electronic
noise and shaping contribution changes were predicted; the axial nonuniformities become
more prominent because the shorter shaping time creates different amounts of ballistic
deficit along the detector’s axis, depending upon the local electric field strength. The
photopeak alignment is less successful in this case because the anode sum is also shaped
with a 12 μs time constant, which introduces severe ballistic deficit into the shaped anode
sum amplitude and compresses the calculated radii into 5 radial bins instead of 6. There
is more photopeak centroid variation within each radial bin, an effect that cannot be
compensated unless the number of bins is increased. A summary of the simulation
results with a 12 μs shaping time are compared to the 20 μs case in Table 6.3; only
parameters that differ from those in Table 6.2 are listed.
127
Figure 6.11. A comparison of the peak broadening terms for two different shaping times.
Table 6.3. A comparison of FWHM contributions for two shaping times.
Process FWHM (channels), 12 μs FWHM (channels), 20 μs
Shaping, geometry 6.99 3.34
Shaping, timing 1.64 2.50
Electronic noise 22.29 24.74
Axial nonuniformity 12.29 10.37
Peak alignment (6.94i) (8.03i)
128
6.2 The Effects of Multiple-Site Events and Interaction Location
It is reasonable to investigate how many interactions recorded in the energy
spectrum are due to single-, double-, triple-, and quadruple-site events (the probability of
more than four event sites in a single event sequence is quite small). Multiple-site events
will contribute to the time-of-arrival contribution to the total peak FWHM, and although
this effect does not appear to be significant from the above simulations, larger volumes or
higher gas densities would increase the multiple-site contributions. Besides the potential
for photopeak broadening, multiple-site events will not register correctly in the radially-
separated spectra, but instead will appear at the product of modified individual event
radii. To prove this point, consider an N -site deposition; nq electrons are ionized at
interaction location n , which has initial difference and sum weighting potentials ,diff inϕ
and ,sum inϕ , respectively. The superscripts f and i refer to each charge cloud’s final and
initial locations. In addition, assume ideal detector behavior, with proper collection of all
electrons at the collecting anode and negligible difference between the two anodes’
weighting potentials throughout the sensitive volume. The measured signals are
, ,
1 1 0
1
, ,
1 1
,
1 1
Ndiff f diff i
diff n n nn
N
nn
Nsum f sum i
sum n n nn
N Nsum i
n n nn n
Q q
q
Q q
q q
ϕ ϕ
ϕ ϕ
ϕ
=
=
=
= =
⎛ ⎞Δ = − −⎜ ⎟⎜ ⎟
⎝ ⎠
−
⎛ ⎞Δ = − −⎜ ⎟⎜ ⎟
⎝ ⎠
≅ − +
∑
∑
∑
∑ ∑
(6.1)
Equation (5.5) can be used to calculate the normalized interaction radius, 0 catr R :
129
,
1 1 1
,
1 1
,
1
1
0
1
1
sumdiff
N N Nsum i
n n n nn n n
N Nsum i
n n nn n
Nsum i
n n nn
cat
cat an
q q q
cat
an
q q
cat
an
q qNcat
n an
r RR R
RR
RR
RR
ϕ
ϕ
ϕ
= = =
= =
=
⎛ ⎞Δ −⎜ ⎟Δ⎝ ⎠
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟− + − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
−
−
=
⎛ ⎞= ⎜ ⎟
⎝ ⎠
∑ ∑ ∑⎛ ⎞= ⎜ ⎟
⎝ ⎠
∑ ∑⎛ ⎞= ⎜ ⎟
⎝ ⎠
∑⎛ ⎞= ⎜ ⎟
⎝ ⎠∏
(6.2)
For a single-site interaction the radius expressed in terms of the weighting potential sum
is
,
0
sum i
cat
cat an
r RR R
ϕ−⎛ ⎞
= ⎜ ⎟⎝ ⎠
(6.3)
Thus, comparing equations (6.2) and (6.3) shows that, in theory, the multiple-site
deposition causes the calculated radius to be the product of individual radii with energy-
weighted exponential arguments.
For some data in Chapter 5, the radial parameter is not calculated using Equation
(5.5), but instead the data is binned as a function of the ratio of anode sum and difference
signals. In this case a multiple-site event registers in the bin containing the weighted
average of the individual bins: this can be shown directly from Equations (6.1):
,
1 1
1
,
1
1
1
N Nsum i
n n nsum n n
Ndiff
nn
Nsum i
n nn
N
nn
q qQQ q
q
q
ϕ
ϕ
= =
=
=
=
− +Δ
=Δ −
= −
∑ ∑
∑
∑
∑
(6.4)
130
Because multiple-deposition events will in general register at incorrect radii, they
cannot be compensated correctly during photopeak alignment. If it is known that these
events are relatively prominent, it may be useful to implement a readout system that can
properly compensate each deposition, such as pixellated anodes with depth sensing
capabilities [65, 101].
To properly investigate the prominence of multiple-site events, let us run the
Geant4 simulations described in Section 6.1.2 (20 μs shaping time constants) while
recording the number of energy depositions for each primary photon; then, event number
statistics and energy spectra can be created for each recorded event number. Let us lump
all event numbers of 5 or higher together into a single tally, and let us also create separate
tallies of the number of full-energy depositions and the number of events occurring in the
photopeak, defined as the region between channels 550 and 700 in the energy spectra.
These two tallies are not necessarily the same, since energy can be deposited in regions of
low sensitivity, therefore resulting in a significantly-lower recorded pulse amplitude. The
results from this simulation are specific only to the source defined in the model: changing
the gamma-ray energy will certainly change the relative prominence of multiple-site
events.
In Figure 6.12, it is obvious that single-site events are by far the most common of
all event numbers; beyond three-site events, the contribution to the total energy spectrum
is negligible, which is why this data is not displayed. There are also a few other effects
that are of interest from the spectrum: the photopeak seems to broaden somewhat as the
number of recorded interactions increases, likely caused by the variation of the detector
response at each interaction site; the photopeak centroid decreases as the number of
interaction sites increases, presumably created by drift-time differences causing the post-
shaping pulse-height deficit discussed earlier in this chapter; and the photopeak becomes
more prominent in comparison to the Compton continuum, which makes sense since full-
energy depositions are more likely with multiple interactions. Table 6.4 lists the
numerical values obtained for photopeak centroid; FWHM (without peak alignment); the
peak-to-total count ratio, where the peak counts are those between channels 550 and 700;
and the total number of counts in the spectrum. The photopeaks for the 4- and 5+-site
spectra are so poorly defined that photopeak information was not obtained. It seems that
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as a rule of thumb, increasing the number of interaction locations by one decreases the
probability of occurrence by roughly an order of magnitude.
Table 6.4. Comparing spectral features as a function of the number of interaction sites.
Event Number Peak Centroid Peak FWHM Total Counts Pk/Total Ratio
1 625.9 27.36 2.378x106 0.067
2 621.9 28.30 4.081x105 0.167
3 618.3 27.88 6.091x104 0.258
4 - - 7.231x103 0.331
5+ - - 0.703x103 0.358
Figure 6.12. Simulated 137Cs energy spectra plotted as a function of the number of interaction sites. The data for 4 and 5+ interactions are not shown because those spectra cannot be seen on this vertical scale to the small number of recorded counts.
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(a) (b)
(c) (d)
Figure 6.13. A comparison of simulated 137Cs radially-separated energy spectra for (a) 1-site, (b) 2-site, (c) 3-site, and (d) 4-site events.
The radially-separated energy spectra for all event sequence types—except those
with 5 or more sites, which are too rare for a spectrum to have meaningful statistics—are
shown in Figure 6.13. For comparison, the corresponding all-event radial spectrum has
been presented already as Figure 5.4. These spectra show that as the event number
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increases, the most probable measured radius decreases, from bin 9 to 8 and finally to 7.
Let us apply the photopeak alignment algorithm developed for the simulated all-event
energy spectrum to each individual interaction-number spectrum to determine the effect
of improper radius measurement on the ability to correct photopeak misalignment. The
results from this study are presented in Table 6.5, with the spectra plotted in Figure 6.14;
it is clear that the single-site interaction spectrum has improved energy resolution after
the photopeak alignment procedure, but multiple-site events actually exhibit degraded
resolution due to the compensation. This confirms the prediction that multiple-site events
would not be properly compensated.
Figure 6.14. Simulated 137Cs spectra sorted by the number of interaction sites. Photopeak alignment has been applied to each spectrum individually.
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Table 6.5. Comparing the simulated energy resolution before and after peak alignment.
Number of interaction sites Resolution (FWHM at 662 keV) before peak alignment
Resolution (FWHM at 662 keV) after peak alignment
1 4.4% 4.2%
2 4.6% 4.7%
3 4.5% 5.1%
Table 6.6. Comparing the prominence of event sequences for 137Cs gamma-ray detection.
Number of interaction sites Full-energy depositions Recorded photopeak events
1 64.3% 64.6%
2 28.2% 27.9%
3 6.5% 6.5%
4 1.0% 1.0%
5+ 0.1% 0.1%
Total number of events 349,828 238,970 (68.3%)
Table 6.6 shows the results of the full-energy event and photopeak tally
breakdowns. The two tallies are in numerical agreement when statistical uncertainties are
considered, meaning it is no more or less likely for a single-site full-energy event to
appear in the photopeak than a three-site full-energy event. From this table, it is apparent
that beyond three interactions the probability of an event sequence contributing to the
photopeak is negligible. The last row totals the number of each tally type over all event
sequences, and the results indicate that, for 662-keV gamma rays, when the full energy is
deposited in this HPXe detector, 68.3% of these events will be recorded in the photopeak.
The rest will be spread out to higher energies (due to weighting potential) or lower
energies (weighting potential or low electric field strength). This result would seem to
indicate that most of these events are shifted to low energies due to the local electric field
strength, as about 1/3rd of the total gas volume lies in regions suspected to suffer low
field strengths—primarily the end gas regions, but also the core of the anode structure.
The end regions of the detector are known to have weak fields from the Maxwell
3D simulations: the reason this is so is because the cathode does not extend into these
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regions, so the outer boundary is held at ground potential instead of -4000 V. A plot
showing the field magnitude in the XZ plane is shown in Figure 6.15. The scale is
chosen such that the dark blue corresponds to regions where electrons drift at less than
the saturation velocity of 1 mm/μs [48]. The field in the central gas space is uniform and
sufficient to maximize the drift velocity, except inside the core of the anode structure.
The very high fields at the outer wall are not in the gas volume, but are instead in the
small gap between the biased cathode and the grounded pressure vessel. Outside of this
central region, though, most of the chamber suffers weak field strength except for very
near the anode wire ends. One final point of interest is that the field strength can be seen
to increase near the cathode at the ends of the central volume, explaining the
recombination variation observed in Figure 6.7.
Figure 6.15. Maxwell 3D simulation results displaying the electric field strength on the XZ plane when the cathode and collecting anode are biased to -4000 V and +1400 V, respectively.
136
Because the fields in the end regions are weak, energy deposited in these spaces
are expected to suffer from significant charge recombination and ballistic deficit. To test
this theory, the previous Geant4 137Cs point source simulation used to study event
sequences was also used to create two more spectra: one with all events included, and one
that omitted events that deposited some energy in the end regions. The results are shown
as Figure 6.16. It is clear that, as predicted, the photopeak region is not affected
significantly by end-event exclusion. On the other hand, for energies in the Compton gap
and below, the number of counts registered in each channel decreases when the end
events are discarded. This effect can be clearly observed in collimation experiments; see,
for example, Figures 4.13 and 4.14.
Figure 6.16. Simulation results of 137Cs spectra for all events (black) and excluding events that contain at least some energy deposited in the end gas spaces (red).
137
6.3 Structural Material Effects on the Energy Spectrum
To clearly identify multiple photopeaks in a single energy spectrum, the
prominence of Compton continua should be reduced as much as possible. This is partly a
function of the cross sections of the detection medium, but it also depends upon the
amount of scattering off of surrounding materials. In the case of HPXe chambers, the gas
comprises only a small fraction of the total detector mass: 330 g of 4.960 kg for detector
HPXe1, 295 g out of 4.670 kg for HPXe2. The significant mass attributed to structural
materials is expected to contribute heavily to the Compton continuum in the recorded
energy spectra. To test this hypothesis, a Geant4 simulation was set up to investigate the
effects of the surrounding structural materials. A 137Cs point source aligned at the
detector’s central axis was simulated; this is important to state because the source
positioning will have an important effect on the results of this simulation. In this case,
the source is positioned such that the gamma rays will generally be passing through the
minimum amount of structural material possible to reach the Xe fill gas, so it is nearly a
best-case scenario. If the source were irradiating the detector from one end, the thickness
of steel and Macor the gammas would pass through before reaching the sensitive volume
would increase dramatically, thus increasing the fraction of photons reaching the HPXe
gas having been scattered into lower energies.
In this simulation, four cases have been studied: one with the normal structural
materials in the appropriate dimensions; one with the steel pressure vessel replaced by a
vacuum; one with the Macor internal structures replaced by a vacuum; and one with both
the steel and the Macor replaced by vacuum. By substituting the normal materials with
vacuum, the material effects can be isolated without geometry changes. The final
simulation case effectively is modeling only the Xe gas, so it is a best-case scenario as far
as the peak-to-Compton count ratio is concerned. The results are shown in Figure 6.17.
It is evident from this simulation that the maximum photopeak height is reduced by about
1/7th due to the presence of scattering materials, while these materials also more than
double the height of the backscatter peak. Again, since the modeled point source was
placed in the most favorable geometry for this simulation, the spectral changes are
expected to be even more dramatic for other source locations.
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Figure 6.17. The effect of replacing structural materials in the HPXe detector with vacuum on simulated 137Cs energy spectra.
6.4 Summary of Simulation Results
The simulations quantifying physical process contributions to spectral degradation
point to the electronic noise as the principal problem regarding peak width. Other non-
negligible factors include the axial nonuniformity of the electric and weighting fields, and
the variation in charge recombination throughout the chamber. Assuming perfect
compensation of every event for recombination and axial nonuniformity, and also
assuming the electronic noise can be reduced to nearly nothing, the energy floor is
expected to be about 0.9% FWHM at 662 keV for the 20 μs shaping time example, or
1.3% for the 12 μs example. Obviously these assumptions are impossible to realize in
experiments, but they do show the absolute performance limitations.
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The analysis also show that multiple-site events account for about 35% of
photopeak counts for the simulated 137Cs point source. These events were shown to
degrade the energy spectrum via a combination of photopeak centroid shifts and peak
broadening. To make matters worse, these multiple-site events generally are registered at
the wrong event radius, and it is not possible to employ photopeak alignment to improve
the energy resolution of multiple-site events—in fact, the alignment procedure further
degrades the spectrum of multiple-site events.
The two paragraphs above point toward the necessity of three-dimensional
position-sensing, which would be able to identify multiple-site events and provide the
capabilities to perform three-dimensional corrections to the measured pulse amplitudes,
therefore pushing the energy resolution as close to the limit as possible. In addition,
these geometries generally have small capacitances that are conducive to low-noise
measurements.
A final suggestion to increase the usefulness of HPXe detectors is to reduce the
structural material as much as possible. The structural material decreases the measured
photopeak efficiency, making it more challenging to pick out a low-activity, low-energy
source when it is engulfed in the Compton continuum of a high-energy gamma ray. This
problem is being approached by Ulin et al. [48], who are constructing pressure vessels
out of very thin steel tubes coated with carbon fiber composites to achieve sufficient
container strength while minimizing the pressure vessel’s mass.
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CHAPTER 7
IMPROVING ENERGY RESOLUTION WITH COOLING ADMIXTURES
7.1 Increasing Drift Velocity with Cooling Admixtures
7.1.1 The Need for Larger Electron Drift Velocities
One common problem in HPXe ionization chambers is the slow drift of electrons
in pure Xe, which saturates near 1 mm/μs, creating the need for long shaping time
constants [48]. As observed in this chamber, the minimum electronic noise is measured
near 6 μs, but the best energy resolution is obtained using a 12 μs shaping time. The
longer time constant is necessary to balance electronic noise with ballistic deficit effects.
As suspected from previous experiments and from the simulation results in Chapter 6,
electronic noise is the dominant contribution to the measured photopeak width, and ideal
performance would be realized if the shaping time could somehow be reduced to take
advantage of the lower electronic noise contributions present at smaller shaping time
constants. This could occur if the electron drift velocity could be increased in the gas,
thereby reducing the ballistic deficit problem at shorter shaping times.
7.1.2 The Effects of Cooling Admixtures
In HPXe detectors, cooling admixtures are small amounts of additives mixed into
the purified Xe gas that have the effect of increasing electron drift speeds through the
chamber. Typically, small concentrations of H2 gas—less than 1% of the molecules in
the mixture—are used to achieve these results, but other admixtures have been
investigated, including He and CH4 [102].
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These admixtures absorb more energy than Xe atoms would in collisions with
drifting electrons, thereby reducing the distribution of electron energies in the electron
cloud; since the average electron energy is given by 32 kT , reducing electron energy can
be thought of as “cooling” [103, 104]. Let us refer to the mean energy lost in a collision
as ( )εΛ , where ε is the electron’s energy. Then
( ) ( ).2
inelm f
Mε εΛ = + ; (7.1)
in this equation, the first term on the right-hand side represents the loss due to elastic
scattering of electrons of mass m interacting with atoms of mass M . The second term
represents inelastic scattering, and it is this term that the cooling admixtures increase
substantially due to their low-energy excitation, vibrational, or rotational states [103].
Figure 7.1. The momentum-transfer cross section for Xe (reprinted from [98]). Cooling admixtures shift the electron energies close to the deep minimum near 0.7 eV.
Although it seems counterintuitive that reducing electron energy can increase the
drift velocity, the decrease in the electron thermal energy distribution coincides with a
very large decrease in the momentum-transfer cross section: see Figure 7.1. This means
the electrons can drift longer distances between collisions with gas molecules, with fewer
142
scatters that slow the electrons and alter their direction. The electrons therefore are able
to follow the field lines more closely, reaching the anode having undergone fewer
scatters, on average. This reduces not only the drift time, but also the diffusion of
particles, resulting in a tighter distribution of electron collection times. Quantitatively,
the drift velocity W and the diffusion coefficient D are represented by the equations
( ) ( )
( ) 1
2 13 3
13
l v dl ve em v m dv
l vD
v
−
= +
=
E EW
(7.2)
with ( )l v the mean free path between collisions, v the thermal velocity of the electrons,
e the fundamental charge, and E the local electric field [104]. The parameter in
brackets is simply the inverse collision frequency; by increasing ( )l v and decreasing v ,
the collision frequency decreases, thereby increasing the drift velocity and decreasing the
diffusion coefficient. If one balances the energy gained from the field and lost via
collisions in a given time increment, then it is not difficult to solve for v and find
3
W El
ElD
∝ Λ
∝Λ
(7.3)
Because hydrogen is the most efficient cooling admixture, it is the most
frequently used in HPXe detectors. The amount of improvement in the drift velocity is a
function of not only the amount of hydrogen added, but also the local electric field
strength, as shown in the published data of Figure 7.2 [48]. This data shows that, with
just a modest amount of hydrogen added to the xenon, the drift velocity can be increased
by up to a factor of ten. It is important to note, though, that for low fields the electron
drift velocity can actually be slower in the xenon/hydrogen mixture than in pure Xe.
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Figure 7.2. Electron drift speed (mm/μs) with different amounts of H2 as a function of electric field strength. The gas density is held constant at 0.6 g/cm3. The H2 concentration for data series 1 to 6 is: 0.0%, 0.2%, 0.3%, 0.5%, 0.7%, 1.0%.
7.1.3 Effects of Cooling Admixtures on Other Detector Properties
It is reasonable to wonder whether the cooling admixtures affect other detector
properties besides the drift velocity. The changes in detection efficiency and interaction
cross-sections are negligible due to the very small admixture concentration.
Recombination along δ-ray tracks becomes more severe due to the more efficient
thermalization of electrons, meaning they do not escape the charge cloud as quickly and
are more susceptible to recombination. The mean ionization energy, w , for a Xe+H2
mixture is found to be a function of both gas density and H2 concentration, but generally
the impact of the cooling admixture is fairly small; see Figure 7.3, reprinted from Ulin
[48]. Generally HPXe detectors are filled to at most 0.6 g/cm3, so departure from the
pure-Xe w is minimal.
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Figure 7.3. A comparison of w in doped HPXe, where W0=21.9 eV. The H2 concentration is: 0.0% (triangles), 0.1% (filled circles), 0.5% (open circles), and 0.7% (inverted triangles). Data reprinted from Ulin [48].
7.2 Simulation Results for Xenon+Hydrogen Mixtures
The simulation package developed in Chapter 6 to incorporate and quantify the
effects of important physical processes has been utilized to simulate the expected energy
spectra for the new mixture of Xe and H2 gases. The only real differences between this
simulation and those in Chapter 6 are:
• the gas composition is changed from 100% Xe to 0.2% H2, 99.8% Xe;
• the shaping time has been shortened to 10 μs;
• the system ENC has been reduced to 381 electrons, corresponding to the new
shaping time; and
• axial field nonuniformity is not considered at this time.
To choose a proper shaping time constant, first the detector response is modeled
for a variety of shaping times; these simulations are identical in nature to those presented
145
in Figures 5.2 and 5.3. The only difference between this simulation and the previous case
is that gas properties for a 0.2% H2 composition have been used. The results are shown
in Figure 7.4, and indicate that even for the shortest shaping time tried, ballistic deficit is
no longer of concern. Choosing a time constant of 10 μs seems to do well for minimizing
ballistic deficit appearing in the calculated event radius.
Figure 7.4. A comparison of detector responses for full-energy events when several shaping time constants are used. (Left) The energy spectrum. (Right) The calculated vs. actual event radius.
Now let us consider a full simulation using a 10-μs shaping time, similar to the
studies of Section 6.1.2. The only significant difference in the new results seems to be
that recombination is more prominent, which coincides with the expectations delineated
in Section 7.1.3. A plot comparing the recombination for the Xe+H2 gas mixture vs. that
for pure gas with a 20-μs shaping time constant is shown in Figure 7.5. The FWHM
contribution due to recombination in this case is substantial, 11.28 keV, compared to 6.77
keV in the pure Xe case. Fortunately, most of this is expected to be compensated by
photopeak alignment, but that would not be possible if the HPXe detector were incapable
of radial position sensing. Figure 7.6 compares these simulated spectra before and after
photopeak alignment. The simulation’s energy resolution improves from 3.7% to 3.3%
FWHM at 662 keV via the alignment process; this result seems outstanding, but it is
146
important to remember that the detrimental effects of axial field nonuniformities are not
currently being considered.
Figure 7.5. A comparison of the simulated distribution of pulse amplitudes after recombination for two gas compositions. The distribution of pulse amplitudes after weighting potential consideration is shown for comparison.
147
Figure 7.6. A comparison of simulated 137Cs spectra using a Xe+H2 gas mixture before and after photopeak alignment.
7.3 Hydrogen Addition and Gas Filling
7.3.1 Choosing the Optimal Hydrogen Concentration
To choose the optimal hydrogen concentration for a particular detector, two
parameters must be known in advance: the desired gas density and the expected range of
electric field magnitudes in the sensitive region of the chamber. Let us keep the same
density used in the pure Xe filling, 0.3 g/cm3. Let us also assume that the chamber will
be biased to a maximum of -4500 V on the cathode, +1400 V on the collecting anode.
This means the electric field strength near the cathode will be about 750 V/cm. After
appropriately scaling the data in Figure 7.2 for density differences, a H2 concentration of
0.2% was found to be appropriate. This corresponds to a concentration for which the
148
electron drift velocity in the gas mixture will be equal to, or greater than, it would be for
pure Xe throughout the entire sensitive volume of the detector.
7.3.2 Gas Mixing, Purification, and Filling
The gas purification system and filling procedure is largely the same as described
in Chapter 4. The detectors were baked for several days to accelerate outgassing of
impurities from the detector internals; an ultra-high vacuum was established to remove
these contaminants from the system. The xenon was purified in the spark purifier for
several weeks prior to filling. When the Xe purification and detector baking were
sufficient, the H2 gas was added to the spark purifier. This step could not be performed
far in advance of the actual filling, because the spark purifier treats H2 as a contaminant
and slowly removes it from the Xe gas, thus reducing the H2 concentration over time.
To obtain the proper amount of H2 in the mixture, first the mass of Xe in the spark
purifier was estimated using the known vessel volume and the measured pressure at room
temperature. This allowed direct calculation of the amount of H2 necessary to achieve
the desired concentration in the final gas mixture. The spark purifier was then cooled in a
liquid N2 bath: at 77 K, Xe solidifies in the bottom of the vessel, but H2 remains a gas, so
measuring the vessel’s gas space pressure provides a simple way to measure the number
of H2 molecules present. It is always imperative to reduce the concentration of
electronegative impurities as much as possible, since they will scavenge drifting electrons
to form negative ions, so H2 was passed through a getter to reduce H2O, O2, CO, and CO2
impurities to less than 1 ppb [105]. After the H2 was purified, it was directed into the
spark purifier, which was filled until a predetermined H2 overpressure was measured. At
that point, the correct amount of H2 had been added, and the system was sealed and
allowed to warm to room temperature. After sufficient warming and gas mixing, the H2
concentration was verified independently by passing small amounts of the gas mixture
into a mass spectrometer and measuring the ratio of H2 to Xe. Once the H2 concentration
was verified, the detector was filled to a pressure corresponding to the desired density,
sealed, and removed from the filling system.
The final measured gas densities and the H2 concentrations measured with the
mass spectrometer are listed in Table 7.1. The gas filling came very close to the desired
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H2 concentration and gas density of 0.2% and 0.3 g/cm3. Detector HPXe2 was filled
about one week after HPXe1 without further addition of H2; the scavenging of H2 in the
spark purifier can be verified in the H2 concentration measured just prior to filling.
Table 7.1. Gas properties of the filled HPXe detectors.
Detector Added gas mass Final gas density H2 concentration
HPXe1 375 g 0.320 g/cm3 0.23 %
HPXe2 390 g 0.332 g/cm3 0.16 %
7.4 Gamma-Ray Measurements with Xenon+Hydrogen Gas Mixtures
7.4.1 Initial Experiments
After filling, the detectors were tested using the same peak-hold system described
in Chapter 5. The cathode bias was set at -4800 V for detector HPXe2 and -5000 V for
HPXe1. The power supply was able to deliver -5000 V, but detector HPXe2 was
constrained to lower bias due to arcing at -5000 V. The collecting anodes on the
detectors were held at +1400 V for HPXe2 and +1600 V for HPXe1, conditions known
from previous testing to be near the point where arcing occurs.
At these biases, pulse waveforms prior to shaping were studied to determine the
effect of the gas filling. The anode difference signal was split using a tee: one branch
was sent to the oscilloscope, the other to a shaping amplifier. The shaping amplifier
output was directed to the oscilloscope and used to trigger the system; a trigger level was
set corresponding to a pulse amplitude in the 137Cs Compton gap, so presumably only
full-energy events were observed. Anode difference rise times in the 2-3 μs range were
observed, and the rise time of the anode sum signal was always less than 20 μs. If these
values are compared to the measurements for pure Xe in Chapter 4, it is evident that the
new gas mixture approximately doubled the electron drift velocity throughout the
sensitive volume, which is exactly the result desired from the Xe+H2 mixture.
Once a detector was properly biased, the appropriate shaping time for the anode
difference signal needed to be established. To properly choose this filter, a series of
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collimated 137Cs measurements was made with only the shaping time differing from one
measurement to the next. The results for detector HPXe2 are listed in Table 7.2; of these
choices, clearly the Canberra 2026 amplifier with a Gaussian filter and a 12 μs shaping
time was the optimal choice. Evidently ballistic deficit must still be an important factor,
as the optimal shaping time still does not correspond to the electronic noise minimum.
Table 7.2. A summary of collimated 137Cs measurements during the shaping time study.
Shaping time Filter type Amplifier Photopeak resolution
Electronic noise limit
6 μs Gaussian ORTEC 672 6.8% 3.6%
8 μs Gaussian Canberra 243 5.8% 3.5%
10 μs Gaussian ORTEC 672 5.9% 3.7%
12 μs Gaussian Canberra 2026 5.4% 4.0%
12 μs Triangular Canberra 2026 5.7% 3.9%
16 μs Gaussian Canberra 243 5.8% 4.0%
7.4.2 Radial Position-Sensing Experiments
After choosing the optimal shaping filter for the anode difference signal, radial
sensing data collection commenced. For this measurement the anode sum was shaped
with a 16 μs time constant to minimize any effect of ballistic deficit. The radial position
was estimated using the ratio of the anode sum to the anode difference signal; the range
of expected ratios was divided into 10 bins, based upon the expected position uncertainty,
and extra bins were placed just outside this range to register unexpectedly high or
negative radial coordinates. For the remainder of this section, data presented will be for
detector HPXe2 unless noted otherwise.
Measurements were made in 30-minute intervals. First a measurement was made
with a 137Cs source collimated to irradiate only the central plane of the detector with a
beam about ¼-inch wide. A test pulse was injected to quantify electronic noise. After
this measurement, a background spectrum was recorded for an equivalent counting time.
Data processing included background stripping followed by photopeak alignment as a
151
function of measured radial bin. Photopeak alignment was found to be much more
important in the Xe+H2 gas mixture than in pure Xe experiments with very similar
biasing and shaping, as demonstrated by the measured photopeak centroid at each radial
bin, presented for both detector gas fills in Table 7.3. This measured effect was predicted
by the simulations in Section 7.2. Figure 7.7, which separates the background-stripped
energy spectra as a function of measured radial bin, shows this photopeak shift clearly as
a function of radial coordinate. The degradation of the weighting potential uniformity
near the anodes (radii 2, 3, and 4) can clearly be seen in this figure.
Figure 7.7. Experimental 137Cs energy spectra as a function of radial bin. The peaks near channel 800 in radial bins 9 and 10 are from an injected test pulse.
Table 7.3. A comparison of measured photopeak centroids as a function of radial bin.
Gas Rad 0
Rad 1
Rad 2
Rad 3
Rad 4
Rad 5
Rad 6
Rad 7
Rad 8
Rad 9
Rad 10
Rad 11
Pure Xe - - - 639 636 635 629 - - - - -
Xe+H2 - - - - 593 592 589 585 579 566 - -
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The result of the photopeak alignment is displayed in Figure 7.8, along with the
uncorrected and background-stripped spectra for comparison. It is clear that the
photopeak alignment noticeably reduces the photopeak width. A test pulse appears near
channel 800; the alignment process causes the test peak to be shifted in the final
spectrum. The measured energy resolution of the 137Cs source after photopeak alignment
is 4.2% FWHM, which is a substantial improvement over the best result in pure Xe, 5.5%
FWHM; nearly identical electrode biasing, source geometry, and shaping filter settings
were used in these two experiments. The result for detector HPXe1 showed a similar
improvement, 4.4% FWHM at 662 keV.
Figure 7.8. A collimated 137Cs raw spectrum compared to the background-stripped and the aligned results; a test pulse appears near channel 800. The aligned resolution is 4.2%.
The next experiment investigated the effect of collimation on the detector
performance. Three measurements were made in quick succession with a 137Cs source
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and the system settings identical to the last experiment: a collimated plane irradiating the
center of the detector, a collimated plane displaced by 1.5 inches (this is only 0.5 inch
from the edge of the sensitive volume), and a window collimator that blocks only
gammas which might interact in the end regions of the chamber, which are known to
have non-ideal responses to radiation interactions. The results are presented in Figure
7.9. The important quantitative results are presented in Table 7.4: the measured
photopeak centroid, FWHM, and the peak-to-total count ratio (excluding the test pulse
region). The slight upshift in photopeak centroid predicted by the simulations in Chapter
6 are observed here, along with the corresponding broadening as the beam moves closer
to the end of the detector. Both of these effects were explained by recombination
variation throughout the chamber. The more prominent backscatter peak in the displaced
plane and window collimation data is likely due to the increased probability of gammas
backscattering off of the thick Macor and steel plates near the end of the pressure vessel.
Figure 7.9. A comparison of 137Cs spectra with three different collimator configurations.
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Table 7.4. A comparison of spectral parameters in the 137Cs collimation experiment.
Collimation Photopeak centroid Photopeak FWHM Peak/total ratio
Center plane 661.7 30.3 0.0858
Displaced plane 666.2 32.7 0.0724
Window 663.8 31.4 0.0725
(a) (b)
(c) (d)
Figure 7.10. Collimated source spectra after photopeak alignment: (a) 133Ba, (b) 57Co, (c) 60Co, (d) 54Mn. All spectra were recorded using the same system settings and counting time.
155
A final test performed with the HPXe detector was to re-check the measured
photopeak linearity and energy resolution trends with several point sources in the central-
plane collimation geometry. The system settings were held constant from the previous
experiments. The spectra are shown in Figure 7.10 after background stripping and
photopeak alignment. The improved energy resolution immediately becomes clear when
comparing the 60Co spectrum to the multiple-source spectrum in Figure 4.17: the 1173
keV peak is clearly distinguishable from the 1332 keV line’s Compton edge with the
improved energy resolution, whereas with pure Xe this measurement could not clearly
distinguish the two features.
The measured centroid location and FWHM are listed for each published gamma
line in Table 7.5 [93]. The linearity is displayed in Figure 7.11, and a comparison of the
intrinsic measured energy resolution—i.e., after electronic noise contributions are
subtracted out in quadrature—to a line with the theoretical slope of -½ predicted by Fano
carrier statistics is presented in Figure 7.12 [8]. In each plot, the error bars extend a
distance of one standard deviation in each direction from the measured centroid or
FWHM value, respectively [106].
Table 7.5. Collimated source photopeak data for detector HPXe2.
Source Gamma Energy (keV) / Intensity (%)
Measured Centroid (channels)
Measured FWHM (channels)
133Ba 81.0 keV (34.1%) 80.9 25.0 133Ba 302.9 keV (18.3%) 306.5 15.2 133Ba 356.0 keV (62.1%) 357.1 24.7 57Co 122.1 keV (85.6%) 120.8 23.3 60Co 1173.2 keV (99.9%) 1168.6 26.7 60Co 1332.5 keV (100.0%) 1324.7 39.5 137Cs 661.7 keV (85.1%) 661.7 30.1 54Mn 834.8 keV (100.0%) 836.4 30.5
156
Figure 7.11. A plot of the experimentally-measured photopeak centroids vs. the published gamma-ray energy. The red line represents a linear least-squares fit.
Figure 7.12. A plot of the natural logarithm of the measured intrinsic resolution as a function of the natural logarithm of the normalized gamma-ray energy. The red line shows the theoretical slope if only Fano carrier statistics are important.
157
In Figure 7.11 it is once again true that excellent linearity is measured with this
detector. This result is not necessarily a given, since the amount of charge lost to
recombination depends upon the density of ionizations, and is observed to decrease with
increasing gamma-ray energy [107]. This effect, however, seems to be negligible.
In Figure 7.12 it seems that at high energies the energy resolution roughly follows
the trend predicted if only Fano carrier statistics are significant, although there may be
too few data points to make a strong assertion of this observation. At the lowest energy
(81.0 keV) the measured resolution seems to depart from the Fano statistics significantly;
it should be noted from Figure 7.10(a) that this peak is located very near the
discrimination level, potentially impacting the results, although it is expected that the
discriminator would seem to improve the energy resolution by cutting off the low-energy
side of the distribution. If one considers other physical reasons for the low-energy
departure from Fano statistics, the size of the cloud should be beneficial in terms of
shaped amplitude uniformity, and the dominance of photoelectric absorptions should
reduce broadening due to drift-time differences. One strong possibility is that the electric
field is still fairly weak near the outer wall, as evidenced by the sharp drop in the centroid
measured near the cathode—see Table 7.3. This effect will tend to impact the low-
energy gammas most, since they will not penetrate into the chamber as well as photons of
several-hundred keV energies. The low field strength will cause increased charge
recombination and poorer energy resolution.
158
CHAPTER 8
CONCLUSIONS
8.1 Summary of Pure Xenon Experiments and Simulations
The purpose of this coplanar-anode HPXe ionization chamber was to develop a
viable alternative to gridded detectors that would not be susceptible to microphonics-
induced spectrum degradation. Although no vibrational testing was performed, it seems
that this anode design can be made resistant to microphonics, and is therefore reasonable
to pursue in future designs. The geometry chosen for this design was based upon simple
simulations optimally balancing electronic noise and weighting potential effects.
This detector has fairly low preamplifier input capacitance, measured at 21.9±2.0
pF, most of which is between the two anode sets. This translates into electronic noise
limits that are acceptable but fairly high compared to gridded chambers, estimated to be
about 2.7% FWHM for 662 keV based upon system measurements. This noise floor,
however, was measured with a 6-μs shaping time, which is too short based upon ballistic
deficit effects: in practice, the best performance is measured with a 12-μs shaping time
constant, meaning the electronic noise is measurably larger than the best-case scenario.
Simulations predicted ballistic deficit minimization when using a shaping time near 20
μs, but at that point the electronic noise overwhelms the minimal contribution from
nonideal shaping. Probably the largest challenge facing coplanar-anode implementation
in HPXe is the unusually high electronic noise, which comes about from the combination
of noise from two independent anodes, simplistically increasing the noise FWHM by 2
compared to the noise from a comparable gridded chamber, which has only one readout
channel. This problem is present in CdZnTe coplanar-anode devices as well, but in those
159
detectors the number of ionizations per unit energy deposited is greater by a factor of
about 4.5, therefore reducing the importance of the electronic noise in relative terms.
The measured energy spectra generally exhibit a very prominent Compton
continuum, especially at low energies. This contribution was found to be largely from
two detrimental sources: background radiation and interactions in the end regions of the
detector. The end regions were shown to provide a poor response to gamma-ray
interactions, with no measurable photopeak. For most subsequent experiments, these end
regions were shielded to minimize their detrimental contribution to the measured spectra.
Radial position sensing methods were developed and implemented using the
existing coplanar anodes. This method is among the first position-sensing efforts in
HPXe ionization chambers, and is certainly the most practical to implement. Radial
sensing is useful for detector diagnostics and improving the quality of the measured
energy spectrum. For example, some of the Compton continuum can be rejected using
this information, and photopeak alignment as a function of measured radius can improve
the photopeak energy resolution: for data collected with this detector, the energy
resolution improved from 5.9% to 5.5% FWHM using peak alignment. In the detector
diagnostics realm, data collected using this technique indicates that the weighting
potential uniformity is very poor near the anodes, and an improved anode design could
help to improve the energy resolution contribution from near-anode events.
A detailed study of photopeak broadening contributions found electronic noise to
be the biggest concern, which certainly is verified by experiments. Other important
factors include: the collecting anode bias limitation, which reduces the number of counts
measured in the photopeak; charge recombination, which reduces the measured peak
amplitude while broadening the peak width due to the statistical nature of recombination
and the variation in the field strength; and the axial nonuniformity of the weighting and
electric fields, which causes the photopeak location to shift while its measured width
changes as a function of axial location. All of these effects can be observed
experimentally. The simulations predict a resolution limit of about 1% FWHM at 662
keV; the stipulations are that electronic noise must be so inconsequential as to be
unmeasurable, and that variations in pulse height as a function of position must be fully
compensated on an event-by-event basis.
160
The impact of multiple-interaction events was also studied with simulations.
According to the Geant4 studies, multiple-site events account for about 35% of
photopeak events, so their contribution is non-negligible. Multiple-site events were
found to degrade the measured energy resolution via broader photopeak width and
reduced centroid location. Because these events will often register in the incorrect radius,
photopeak alignment actually degrades the contribution from multiple-site events.
Unfortunately, this effect is difficult, if not impossible, to quantify experimentally.
The large mass of structural material surrounding the detection volume was found
to impact the energy spectrum. By switching volumes between the actual materials and
vacuum, the effect of these materials upon the energy spectrum could easily be studied.
The energy resolution seems to be impacted negligibly, but the number of photopeak
counts is reduced noticeably by the presence of the surrounding scattering sites.
Consequently, the relative importance of the Compton continuum is larger with the
surrounding materials present, doubling the height of the continuum at the backscatter
peak for this particular source energy and location.
8.2 Summary of Xenon+Hydrogen Experiments and Simulations
Cooling admixtures, namely H2, can be added to the Xe fill gas to improve
electron drift speed through the chamber, in some instances by up to a factor of ten.
Simulations predict that a sufficient shaping time for minimizing ballistic deficit is 10 μs
for just 0.2% H2, compared to 20 μs for the pure Xe case. Investigations into the physical
contributions to photopeak broadening show that the major differences between the pure
and doped Xe cases are reduced electronic noise and increased recombination
contributions for Xe+H2, which aligns with expectations and experimental observations.
In experiments with the detectors filled with ~0.2% H2 gas, the measured energy
resolution improved dramatically after photopeak alignment: 4.2% FWHM at 662 keV.
The system was shown to have very good linearity over a range extending from about 80
keV up to 1332 keV. The resolution was tracked as a function of gamma-ray energy, and
for high energies the trend predicted by Fano statistics held true, although at low energies
the data trended toward much higher values. The explanation offered for this observation
is the possibility that the electric field is not strong enough near the cathode, resulting in
161
comparatively large statistical contributions from charge recombination. This effect
would apply mostly to low-energy photons, which will tend to be absorbed more readily
near the cathode than high-energy particles.
8.3 Suggestions for Future Work
This body of work has shown coplanar-anode HPXe chambers to be a promising
concept if the energy resolution can be reduced a little more. Measurements of 4.2%
FWHM at 662 keV are getting close to the typical 3.5% to 4.0% measured in gridded
chambers of similar diameter. To improve the energy resolution as much as is practical,
the possibility of reducing the capacitance between anodes should be studied, as this
parameter affects the electronic noise. Only the number of wires was optimized in this
detector; optimizing the individual wire diameter and the wire offset from the detector’s
central axis may be useful studies. Preamplifier transistor cooling to reduce the series
noise term may make sense.
The axial response of the detector should be made as uniform as possible, as
nonuniformity is a detrimental effect that was measured using source collimation with the
current detector. Proper high-voltage design should be considered to allow the collecting
anode, as well as the cathode, to be biased optimally. In addition, the fraction of gas
volume in regions with poor response, currently more than 25%, should be reduced to
decrease the continuum prominence in the energy spectra.
In other efforts to reduce the continuum prominence, an effort to reduce the
surrounding structural material may be worthwhile. Some literature cites efforts to use a
carbon fiber shell as the pressure vessel, a promising concept. Reducing the thickness of
the structural insulating plates inside the chamber is also a good idea.
Enhancing the position-sensing capability of this system is also a possible future
activity. These capabilities may help to correct for variations in the detector response
throughout the chamber, or may be able to separate single-site events from the resolution-
degrading multiple-site events. It is unclear whether a completely different electrode
concept, such as anode pixellation, would be required for such capabilities.
If coplanar-anode HPXe detectors are to ever be used in real-world applications,
microphonic resistance will need to be proven. Thus, vibrational studies will need to be
162
performed, preferably with both a coplanar-anode and a gridded HPXe chamber to
directly compare results. The current detector did not go through vibrational analysis
during the design or testing phase, so it is unlikely that the current geometry is ideal for
microphonic resilience.
163
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